Research ArticleTopology-BasedAnalysisofOTIS (Swapped)NetworksOKn

andOPn

Hai-Xia Li1 Sarfaraz Ahmad2 and Iftikhar Ahmad 23

1Department of General Education Anhui Xinhua University Hefei 230088 China2Department of Mathematics COMSATS University of Islamabad Lahore Campus Lahore 54000 Pakistan3Department of Mathematics Riphah International University Lahore Campus Lahore 54000 Pakistan

Correspondence should be addressed to Iftikhar Ahmad iffi6301gmailcom

Received 2 September 2019 Accepted 10 October 2019 Published 7 November 2019

Guest Editor Shaohui Wang

Copyright copy 2019 Hai-Xia Li et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In the fields of chemical graph theory topological index is a type of a molecular descriptor that is calculated based on the graph ofa chemical compound In this paper M-polynomial OKn

and OPnnetworks are computed e M-polynomial is rich in in-

formation about degree-based topological indices By applying the basic rules of calculus on M-polynomials the first and secondZagreb indices modified second Zagreb index general Randic index inverse Randic index symmetric division index harmonicindex inverse sum index and augmented Zagreb index are recovered

1 Introduction

Cheminformatics is a new branch of science which relateschemistry mathematics and computer sciences Quantita-tive structure-activity relationship (QSAR) and quantitativestructure-property relationship (QSPR) are the main com-ponents of cheminformatics which are helpful to studyphysicochemical properties of chemical compounds [1ndash3]

A topological index is a numeric quantity associated withthe graph of chemical compound which characterizes itstopology and is invariant under graph automorphism [4ndash6]ere are numerous applications of graph theory in the fieldof structural chemistry e first well-known use of a to-pological index in chemistry was by Wiener in the study ofparaffin boiling points [7ndash9] After that in order to explainphysicochemical properties various topological indices havebeen introduced and studied [10 11]

A computer network is a digital telecommunicationsnetwork which allows nodes to share resources In computernetworks computing devices exchange data with each otherusing connections (data links) between nodes ese data linksare established over cablemedia such aswires or optic cables orwirelessmedia such asWiFi Optical transpose interconnectionsystem (OTIS) networks were initially contrived to give pro-ductive network to new optoelectronic computer models thatprofit by both optical and electronic advancements [12] In

OTIS networks processors are orchestrated into groupsElectronic inter-connects are used between processors withinthe same cluster while optical links are used for interclustercommunication Various algorithms have been produced fordirecting determinationarranging certain numerical calcu-lations Fourier transformation [13] matrix multiplication [14]image processing [15] and so on [16 17] e structure of aninterconnection system can be scientifically modeled bya graph e vertices of this graph are the processor nodes andthe edges are the connections between the processors etopology of a graph decides the manner by which vertices areassociated by edges From the topology of a system certainproperties can be decided e diameter of a graph is themaximum distance between any two vertices of the graph

Definition 1 (OTIS (swapped) network OKn+2) e OTIS

(swapped) network is derived from the graph Ω which isa graph with vertex set V(OΩ) ltg pgt | g p isin V(Ω) andedge set E(OΩ) ltg p1 gt ltg p2 gtg isin V(Ω) (p1 p2)

isin E(Ω)cup(ltg pgt ltg p ggt )| g p isin V(Ω) and gnepe graph of OTIS (swapped) network OKn+2

given inFigure 1 has (n32) + 3n2 + (11n2) + 3 edges and (n + 2)2

vertices

Definition 2 (OTIS (swapped) network OPn) Let Pn be path

of n vertices and OPn be OTIS (swapped) network with basis

HindawiJournal of ChemistryVolume 2019 Article ID 4291943 11 pageshttpsdoiorg10115520194291943

network Pn An OTIS (swapped) network with the basisnetwork P6 is shown in Figure 2

e OTIS (swapped) network graph Opngiven in Fig-

ure 2 has (32)(n2 minus n) edges and n2 verticesIn this paper we aim to compute degree-dependent to-

pological indices of OTIS (swapped) networks OKnand OPn

InSection 2 we give definitions and literature review about TIs InSections 3 and 4 we present themethodology and application ofour results in chemistry respectively Section 5 contains ourmain results and Section 6 concludes our paper

2 Topological Indices (TIs)

TIs are numbers which depend on the molecular graph and arehelpful in deciding the properties of the concerned molecularcompound [18ndash20] We can consider TI as a function whichassigns a real number to each molecular graph and this realnumber is used as a descriptor of the concernedmolecule Fromthe TIs a variety of physical and chemical properties like heat ofevaporation heat of formation boiling point chromatographicretention surface tension and vapor pressure of understudymolecular compound can be identified A TI gives us themathematical language to study a molecular graph ere arethree types of TIs

(1) Degree-based TIs(2) Distance-based TIs(3) Spectrum-based TIs

e first type of TI depends upon the degree of verticessecond one depends upon the distance of vertices and thethird type of TI depends upon the spectrum of graph

21 Zagreb Indices To compute total π-electron energy thefollowing TI is defined

M1(G) 1113944visinV(G)

d2v (1)

But soon it was observed that this index increases withincreases in branching of the skeleton of carbon atoms After10 years Balaban et al wrote a review [21] in which hedeclared M1 and M2 are among the degree-based TIs andnamed them as Zagreb group indices e name Zagrebgroup indices was soon changed to Zagreb indices (ZIs) andnowadays M1 and M2 are abbreviated as first Zagreb indexand second Zagreb index

In 1975 Gutman et al gave a remarkable identity [22]Hence these two indices are among the oldest degree-baseddescriptors and their properties are extensively investigatede mathematical formulae of these indices are

M1(G) 1113944uvisinE(G)

du + dv

M2(G) 1113944uvisinE(G)

dudv(2)

For detailed survey about these indices we refer [23ndash26]Doslic et al [27] gave the idea of augmented ZI whosemathematical formula is

AZI(G) 1113944uvisinE(G)

dudv

du + dv minus 21113888 1113889

3

(3)

22 Randic or Connectivity Index Historically ZIs are thevery first degree-based TIs but these indices were used forcompletely different purposes therefore the first genuinedegree-based TI is the Randic index (RI) which was given in1975 by Randic [28] as

Rminus 12(G) 1113944uvisinE(G)

1dudv

1113968 (4)

Firstly Randic named it as a branching index which wassoon named as connectivity index and nowadays it is calledas RI e RI is the most popular degree-based TI and hasbeen extensively studied by both mathematicians andchemists Randic himself wrote two reviews [29 30] and

ndash5 ndash4 ndash3 ndash2 ndash1 0 1 2 3 4ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

5

Figure 1 OK7

ndash4 ndash3 ndash2 ndash1 0 1 2 3 4ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Figure 2 OP7

2 Journal of Chemistry

many papers and books on this topological invariant arepresent in the literature few of them are [31ndash34] Re-searchers recognized the importance of the Randic index indrug design Bollbas and Erdos famous mathematicians ofthat time investigated some hidden mathematical propertiesof RI [35] after that RI was worth studying and a surge ofpublications began [36ndash39] An unexpected mathematicalquality of the Randic index was discovered recently whichtells us about the relation of this topological invariant withthe normalized Laplacian matrix [38 40 41] e GRIknown as general RI [42] is defined as

Rα(G) 1113944uvisinE(G)

dudv( 1113857α (5)

23 M-Polynomial e mathematical formula of M-poly-nomial is

M(G x y) 1113944uvisinE(G)

xdu y

dv1113872 1113873 (6)

Detailed survey about the definitions of other TIscomputed in this paper and relation of TIs with M-poly-nomial can be seen in [43 44] where

Dx(f(x y)) xz

zx(f(x y))

Dy(f(x y)) yz

zy(f(x y))

Sx(f(x y)) 1113946x

0

f(t y)

tdt

Sy(f(x y)) 1113946y

0

f(x t)

tdt

J(f(x y)) J(f(x x))

Qα(f(x y)) xα(f(x y))

(7)

3 Methodology

To compute the M-polynomial of a graph G we need tocompute the number of vertices and edges in it and dividethe edge set into different classes with respect to the degreesof end vertices From the M-polynomials we can recovermany degree-dependent indices by applying some differ-ential and integral operators

4 Applications in Chemistry

A topological description is used to depict the features of thestudied compounds and indices of graph-theoretical originare used to investigate the correlations between structure andbiological activity [45ndash48] For example the Randic indexdemonstrates great relationship with the physical property ofalkanes e geometric arithmetic index has a similar role asthat of the Randic index e sum-connectivity index is

helpful in guessing the melting point of compounds Zagrebindices are used to calculate π-electron energy

5 Main Results

In this section we compute M-polynomials of understudynetworks and recover nine TIs from these polynomials

51 Results for OPn

Theorem 1 Let OPn be the swapped network then

M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(8)

Proof e OPn network has the following four types ofedges based on the degree of end vertices

E 13 OPn( 1113857 uv isin E OPn( 1113857 du 1 dv 31113864 1113865

E 22 OPn( 1113857 uv isin E OPn( 1113857 du 2 dv 21113864 1113865

E 23 OPn( 1113857 uv isin E OPn( 1113857 du 2 dv 31113864 1113865

E 33 OPn( 1113857 uv isin E OPn( 1113857 du 3 dv 31113864 1113865

(9)

such thatE 13 OPn( 1113857

11138681113868111386811138681113868111386811138681113868 2

E 22 OPn( 11138571113868111386811138681113868

1113868111386811138681113868 3

E 23 OPn( 11138571113868111386811138681113868

1113868111386811138681113868 (6n minus 14)

E 33 OPn( 11138571113868111386811138681113868

1113868111386811138681113868 3(n minus 2)(n minus 3)

21113888 1113889

(10)

Now from the definition of M-polynomial we have

M(G x y) 1113944δleilejleΔ

mijxiy

j

11139441le3

m13x1y3

+ 11139442le2

m22x2y2

+ 11139442le3

m23x2y3

+ 11139443le3

m33x3y3

E 13 OPn( 11138571113868111386811138681113868

1113868111386811138681113868x1y3

+ E 22 OPn( 11138571113868111386811138681113868

1113868111386811138681113868x2y2

+ E 23 OPn( 11138571113868111386811138681113868

1113868111386811138681113868x2y3

+ E 33 OPn( 11138571113868111386811138681113868

1113868111386811138681113868x3y3

2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(11)

Journal of Chemistry 3

Corollary 1 Let OPn be the swapped network then

M1 OPn( 1113857 9n2

minus 15n + 4 (12)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(13)

en

Dx(f(x y)) 2xy3

+ 6x2y2

+ 2(6n minus 14)x2y3

+9(n minus 2)(n minus 3)

21113888 1113889x

3y3

Dy(f(x y)) 6xy3

+ 6x2y2

+ 3(6n minus 14)x2y3

+9(n minus 2)(n minus 3)

21113888 1113889x

3y3

(14)

Now we have

M1 OPn( 1113857 Dxf +Dyf11138681113868111386811138681113868xy1

9n2

minus 15n + 4

(15)

Corollary 2 Let OPn be the swapped network then

M2 OPn( 1113857 27n2

2minus63n

2+ 15 (16)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(17)

en

Dy(f(x y)) 6xy3

+ 6x2y2

+ 3(6n minus 14)x2y3

+9(n minus 2)(n minus 3)

21113888 1113889x

3y3

DxDy(f(x y)) 6xy3

+ 12x2y2

+ 6(6n minus 14)x2y3

+27(n minus 2)(n minus 3)

21113888 1113889x

3y3

(18)

Now we have

M2 OPn( 1113857 DxDyf11138681113868111386811138681113868xy1

27n2

2minus63n

2+ 15

(19)

Corollary 3 Let OPn be the swapped network then

mM2 OPn( 1113857

n2

6+

n

6+

112

(20)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(21)

en

Sy(f(x y)) 23

1113874 1113875xy3

+32

1113874 1113875x2y2

+(6n minus 14)

31113888 1113889x

2y3

+(n minus 2)(n minus 3)

21113888 1113889x

3y3

SxSy(f(x y)) 23

1113874 1113875xy3

+34

1113874 1113875x2y2

+(6n minus 14)

61113888 1113889x

2y3

+(n minus 2)(n minus 3)

61113888 1113889x

3y3

(22)

Now we havem

M2 OPn( 1113857 SxSyf11138681113868111386811138681113868xy1

n2

6+

n

6+

112

(23)

Corollary 4 Let OPn be the swapped network then

Rα OPn( 1113857 6α + 6α2α + 3α2α(6n minus 14) +9α3α(n minus 2)(n minus 3)

2α1113888 1113889

(24)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(25)

4 Journal of Chemistry

en

Dαy(f(x y)) 6αxy

3+ 6αx

2y2

+ 3α(6n minus 14)x2y3

+9α(n minus 2)(n minus 3)

2α1113888 1113889x

3y3

DαxD

αy1113888(f(x y)) 6αxy

3+ 6α2αx

2y2

+ 3α2α(6n minus 14)x2y3

+9α3α(n minus 2)(n minus 3)

2α1113888 1113889x

3y31113889

(26)

Now we have

Rα OPn( 1113857 DαxD

αyf|xy11113872

6α + 6α2α + 3α2α(6n minus 14) +9α3α(n minus 2)(n minus 3)

2α

(27)

Corollary 5 Let OPn be the swapped network then

RRα OPn( 1113857 2α

3α+3α

22α+6n minus 143α2α

+(n minus 2)(n minus 3)

2α3α1113888 1113889

(28)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(29)

en

Sαy(f(x y))

2α

3α1113888 1113889xy

3+

3α

2α1113888 1113889x

2y2

+(6n minus 14)

3α1113888 1113889x

2y3

+(n minus 2)(n minus 3)

2α1113888 1113889x

3y3

SαxS

αy1113888(f(x y))

2α

3α1113888 1113889xy

3+

3α

22α1113888 1113889x

2y2

+(6n minus 14)

3α2α1113888 1113889x

2y3

+(n minus 2)(n minus 3)

2α3α1113888 1113889x

3y31113889

(30)

Now we have

RRα OPn( 1113857 SαxS

αyf|xy11113872

2α

3α+3α

22α+6n minus 143α2α

+(n minus 2)(n minus 3)

2α3α

(31)

Corollary 6 Let OPn be the swapped network then

SSD OPn( 1113857 3n2

minus 2n +13 (32)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(33)

en

SxDy(f(x y)) 6xy3

+ 3x2y2

+3(6n minus 14)

21113888 1113889x

2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

DxSy(f(x y)) 23

1113874 1113875xy3

+ 3x2y2

+2(6n minus 14)

31113888 1113889x

2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(34)

Now we have

SSD OPn( 1113857 DxSyf + SxDyf1113872 111387311138681113868111386811138681113868xy1

3n2

minus 2n +13

(35)

Corollary 7 Let OPn be the swapped network then

H OPn( 1113857 2n2

4minus

n

20minus

120

1113888 1113889 (36)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(37)

en

J(f(x y)) 5x4

+(6n minus 14)x5

+3(n minus 2)(n minus 3)

21113888 1113889x

6

2SxJ(f(x y)) 25x4

4+

(6n minus 14)x5

5+

(n minus 2)(n minus 3)x6

41113888 1113889

(38)

Journal of Chemistry 5

Now we have

H OPn( 1113857 2SxJf1113868111386811138681113868x1

2n2

4minus

n

20minus

120

1113888 1113889

(39)

Corollary 8 Let OPn be the swapped network then

I OPn( 1113857 9n2

4minus81n

20minus1295

(40)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(41)

en

Dy(f(x y)) 6xy3

+ 6x2y2

+ 3(6n minus 14)x2y3

+9(n minus 2)(n minus 3)

21113888 1113889x

3y3

DxDy(f(x y)) 6xy3

+ 12x2y2

+ 6(6n minus 14)x2y3

+27(n minus 2)(n minus 3)

21113888 1113889x

3y3

JDxDy(f(x y)) 18x4

+ 6(6n minus 14)x5

+27(n minus 2)(n minus 3)

21113888 1113889x

6

SxJDxDy(f(x y)) 18x4

4+

6(6n minus 14)

51113888 1113889x

5

+27(n minus 2)(n minus 3)

121113888 1113889x

6

(42)

Now we have

I OPn( 1113857 SxJDxDyf11138681113868111386811138681113868x1

9n2

4minus81n

20minus1295

(43)

Corollary 9 Let OPn be the swapped network then

A OPn( 1113857 729n2

64minus573n

64minus41332

(44)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(45)

enD

3y(f(x y)) 54xy

3+ 24x

2y2

+ 27(6n minus 14)x2y3

+81(n minus 2)(n minus 3)

21113888 1113889x

3y3

D3xD

3y(f(x y)) 54xy

3+ 192x

2y2

+ 216(6n minus 14)x2y3

+2187(n minus 2)(n minus 3)

21113888 1113889x

3y3

JD3xD

3y(f(x y)) 246x

4+ 216(6n minus 14)x

5

+2187(n minus 2)(n minus 3)

21113888 1113889x

6

Qminus 2JD3xD

3y(f(x y)) 246x

2+ 216(6n minus 14)x

3

+2187(n minus 2)(n minus 3)

21113888 1113889x

4

S3xQminus 2JD

3xD

3y(f(x y))

123x2

4+ 8(6n minus 14)x

3

+729(n minus 2)(n minus 3)

641113888 1113889x

4

(46)Now we have

A OPn( 1113857 S3xQminus 2JD

3xD

3yf

11138681113868111386811138681113868x1

729n2

64minus573n

64minus41332

(47)

52 Results for (ORk)

Theorem 2 Let ORk be the swapped network then

M(G x y) nkxky

k+1+

n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(48)

Proof e ORk network has the following two types ofedges based on the degree of end vertices

E kk+1 ORk( 1113857 uv isin E ORk( 1113857 du k dv k + 11113864 1113865

E k+1k+1 ORk( 1113857 uv isin E ORk( 1113857 du k + 1 dv k + 11113864 1113865

(49)

6 Journal of Chemistry

such that

E kk+1 ORk( 11138571113868111386811138681113868

1113868111386811138681113868 nk

E k+1k+1 ORk( 11138571113868111386811138681113868

1113868111386811138681113868 n2(k + 1) minus n(1 + 2k)

2

(50)

Now from the definition of M-polynomial we have

M(G x y) 1113944δleilejleΔ

mijxiy

j

1113944klek+1

mk(k+1)xky

k+1+ 1113944

k+1lek+1m(k+1)(k+1)x

k+1y

k+1

E kk+1 ORk( 11138571113868111386811138681113868

1113868111386811138681113868xky

k+1+ E k+1k+1 ORk( 1113857

11138681113868111386811138681113868111386811138681113868x

k+1y

k+1

nkxky

k+1+

n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(51)

Corollary 10 Let (ORk) be the swapped network then

M1 ORk( 1113857 2nk2

+ nk + n2(k + 1)

2minus n(k + 1)(1 + 2k)

(52)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(53)

en

Dx(f(x y)) nk2x

ky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

Dy(f(x y)) nk(k + 1)xky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(54)

Now we have

M1 ORk( 1113857 Dxf + Dyf11138681113868111386811138681113868xy1

2nk2

+ nk + n2(k + 1)

2minus n(k + 1)(1 + 2k)

(55)

Corollary 11 Let (ORk) be the swapped network then

M2 ORk( 1113857 nk3

+ nk2

+n2(k + 1)3

2minus

n(k + 1)2(1 + 2k)

2

(56)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(57)

en

Dy(f(x y)) nk2x

ky

k+1+ nkx

ky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxDy(f(x y)) nk3x

ky

k+1+ nk

2x

ky

k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(58)

Now we have

M2 ORk( 1113857 DxDyf11138681113868111386811138681113868xy1

nk3

+ nk2

+n2(k + 1)3

2minus

n(k + 1)2(1 + 2k)

2

(59)

Corollary 12 Let (ORk) be the swapped network then

mM2 ORk( 1113857

nk

k2 + k( )+

n2

(2k + 2)minus

n(1 + 2k)

2(k + 1)2 (60)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(61)

en

Sy(f(x y)) nk

(k + 1)1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2(k + 1)1113888 1113889x

k+1y

k+1

SxSy(f(x y)) nk

k2 + k( )1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2(k + 1)21113888 1113889x

k+1y

k+1

(62)

Journal of Chemistry 7

Now we havem

M2 ORk( 1113857 SxSyf11138681113868111386811138681113868xy1

nk

k2 + k( )+

n2

(2k + 2)minus

n(1 + 2k)

2(k + 1)2

(63)

Corollary 13 Let (ORk) be the swapped network then

Rα1113888 ORk( 1113857 nk3α

+ nk2α

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

2α1113888 1113889x

k+1y

k+11113889

(64)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(65)

en

Dαy(f(x y)) nk

2αx

ky

k+1+ nk

αx

ky

k+1

+(k + 1)α n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DαxD

αy1113888(f(x y)) nk

3αx

ky

k+1+ nk

2αx

ky

k+1

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+11113889

(66)

Now we have

Rα ORk( 1113857 DαxD

αyf

11138681113868111386811138681113868xy11113874

nk3α

+ nk2α

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

2α1113888 1113889x

k+1y

k+1

(67)

Corollary 14 Let (ORk) be the swapped network then

RRα ORk( 1113857 n

(k + 1)α+

n2(k + 1) minus n(1 + 2k)

(k + 1)2α2α1113888 11138891113888 1113889

(68)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(69)

en

Sαy(f(x y))

nkα

(k + 1)α1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2α(k + 1)α1113888 1113889x

k+1y

k+1

SαxS

αy1113888(f(x y))

n

(k + 1)α1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2α(k + 1)2α1113888 1113889x

k+1y

k+11113889

(70)

Now we have

RRα ORk( 1113857 SαxS

αyf

11138681113868111386811138681113868xy11113874

n

(k + 1)α+

n2(k + 1) minus n(1 + 2k)

(k + 1)2α2α1113888 1113889

(71)

Corollary 15 Let (ORk) be the swapped network then

SSD ORk( 1113857 nk2

(k + 1)+ nk + n + n

2(k + 1) minus n(1 + 2k)

(72)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(73)

en

SxDy(f(x y)) nkxky

k+1+ nx

ky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxSy(f(x y)) nk2

(k + 1)1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(74)

Now we have

SSD ORk( 1113857 DxSyf + SxDyf1113872 111387311138681113868111386811138681113868xy1

nk2

(k + 1)+ nk + n + n

2(k + 1) minus n(1 + 2k)

(75)

8 Journal of Chemistry

Corollary 16 Let (ORk) be the swapped network then

H ORk( 1113857 2nk

2k + 1+

n2(k + 1)

4k + 4minus

n(1 + 2k)

4k + 41113888 1113889 (76)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(77)

en

J(f(x y)) nkx2k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

2SxJ(f(x y)) 21113888nk

2k + 11113888 1113889x

2k+1

+n2(k + 1)

4k + 41113888 1113889x

2k+2

minusn(1 + 2k)

4k + 41113888 1113889x

2k+21113889

(78)

Now we have

H ORk( 1113857 2SxJf1113868111386811138681113868x1

2nk

2k + 1+

n2(k + 1)

4k + 4minus

n(1 + 2k)

4k + 41113888 1113889

(79)

Corollary 17 Let (ORk) be the swapped network then

I ORk( 1113857 nk3

2k + 1+

nk2

2k + 1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889

(80)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(81)

en

Dy(f(x y)) nk(k + 1)xky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxDy(f(x y)) nk3x

ky

k+1+ nk

2x

ky

k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

JDxDy(f(x y)) nk3x2k+1

+ nk2x2k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

SxJDxDy(f(x y)) nk3

2k + 11113888 1113889x

2k+1+

nk2

2k + 11113888 1113889x

2k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889x

2k+2

(82)

Now we have

I ORk( 1113857 SxJDxDyf11138681113868111386811138681113868x1

nk3

2k + 1+

nk2

2k + 1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889

(83)

Corollary 18 Let (ORk) be the swapped network then

A ORk( 1113857 nk5(k + 1)2

(2k minus 1)3

+nk4(k + 1)2

(2k minus 1)3

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889

(84)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(85)

Journal of Chemistry 9

en

D3y(f(x y)) nk

2(k + 1)

2x

ky

k+1+ nk(k + 1)

2x

ky

k+1

+(k + 1)3 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

D3yD

3y(f(x y)) nk

5(k + 1)

2x

ky

k+1+ nk

4(k + 1)

2x

ky

k+1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

JD3yD

3y(f(x y)) nk

5(k + 1)

2x2k+1

+ nk4(k + 1)

2x2k+1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

Qminus 2JD3yD

3y(f(x y)) nk

5(k + 1)

2x2kminus 1

+ nk4(k + 1)

2x2kminus 1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k

S3xQminus 2JD

3yD

3y(f(x y))

nk5(k + 1)2

(2k minus 1)31113888 1113889x

2kminus 1+

nk4(k + 1)2

(2k minus 1)31113888 1113889x

2kminus 1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889x2k

(86)

Now we have

A ORk( 1113857 S3xQminus 2JD

3yD

3yf

11138681113868111386811138681113868x1

nk5(k + 1)2

(2k minus 1)3+

nk4(k + 1)2

(2k minus 1)3

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889

(87)

6 Conclusion

In this paper our focus is on swapped interconnectionnetworks that allow systematic construction of large scal-able modular and robust parallel architectures whilemaintaining many desirable attributes of the underlyingbasis network comprising its clusters We have computedseveral TIs of underlined networks Firstly we computed M-polynomials of understudy networks and then we recoveredZagreb indices Randic indices and some other indices[49ndash52]

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to this study

Acknowledgments

is study was partially supported by the Natural ScienceFoundation of Anhui Province (no KJ2018A059)

References

[1] W Gao M R Farahani and L Shi ldquoForgotten topologicalindex of some drug structuresrdquo Acta Medica Mediterraneavol 32 no 1 pp 579ndash585 2016

[2] W Gao M K Siddiqui M Imran M K Jamil andM R Farahani ldquoForgotten topological index of chemicalstructure in drugsrdquo Saudi Pharmaceutical Journal vol 24no 3 pp 258ndash264 2016

[3] W Gao W Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 Article ID 3216327 8 pages 2016

[4] M Imran A Asghar and A Q Baig ldquoOn graph invariants ofoxide networkrdquo Engineering and Applied Science Lettersvol 1(2018) no 1 pp 23ndash28 2018

[5] G Liu Z Jia and W Gao ldquoOntology similarity computingbased on stochastic primal dual coordinate techniquerdquo OpenJournal of Mathematical Sciences vol 2(2018) no 1pp 221ndash227 2018

[6] S Noreen and A Mahmood ldquoZagreb polynomials andredefined Zagreb indices for the line graph of carbonnanoconesrdquo Open Journal of Mathematical Analysisvol 2(2018) no 1 pp 66ndash73 2018

[7] S Hayat and M Imran ldquoComputation of topological indicesof certain networksrdquo Applied Mathematics and Computationvol 240 pp 213ndash228 2014

[8] S Hayat and M Imran ldquoComputation of certain topologicalindices of nanotubes covered byrdquo Journal of Computationaland eoretical Nanoscience vol 12 no 4 pp 533ndash541 2015

[9] A Q Baig M Imran and H Ali ldquoOn topological indices ofpoly oxide poly silicate DOX and DSL networksrdquo CanadianJournal of Chemistry vol 93 no 7 pp 730ndash739 2015

[10] Y Bashir A Aslam M Kamran et al ldquoOn forgotten topo-logical indices of some dendrimers structurerdquo Moleculesvol 22 no 6 p 867 2017

[11] A Aslam J L Guirao S Ahmad and W Gao ldquoTopologicalindices of the line graph of subdivision graph of completebipartite graphsrdquo Applied Mathematics amp Information Sci-ences vol 11 no 6 pp 1631ndash1636 2017

[12] G C Marsden P J Marchand P Harvey and S C EsenerldquoOptical transpose interconnection system architecturesrdquoOptics Letters vol 18 no 13 pp 1083ndash1085 1993

[13] A Al-Ayyoub A Awwad K Day and M Ould-KhaoualdquoGeneralized methods for algorithm development on opticalsystemsrdquo e Journal of Supercomputing vol 38 no 2pp 111ndash125 2006

[14] C F Wang and S Sahni ldquoMatrix multiplication on the OTIS-mesh optoelectronic computerrdquo IEEE Transactions onComputers vol 50 no 7 pp 635ndash646 2001

[15] C F Wang and S Sahni ldquoImage processing on the OTIS-mesh optoelectronic computerrdquo IEEE Transactions on Paralleland Distributed Systems vol 11 no 2 pp 97ndash109 2000

[16] C F Wang and S Sahni ldquoBasic operations on the OTIS-meshoptoelectronic computerrdquo IEEE Transactions on Parallel andDistributed Systems vol 9 no 12 pp 1226ndash1236 1998

10 Journal of Chemistry

[17] S Rajasekaran and S Sahni ldquoRandomized routing selectionand sorting on the OTIS-meshrdquo IEEE Transactions on Paralleland Distributed Systems vol 9 no 9 pp 833ndash840 1998

[18] A R Virk M N Jhangeer and M A Rehman ldquoReverseZagreb and reverse hyper-zagreb indices for silicon carbideSi2C3I[rs] and Si2C3II[rs]rdquo Engineering and Applied ScienceLetters vol 1 no 2 pp 37ndash50 2018

[19] N De ldquoComputing reformulated first Zagreb index of somechemical graphs as an application of generalized hierarchicalproduct of graphsrdquo Open Journal of Mathematical Sciencesvol 2 no 1 pp 338ndash350 2018

[20] N De ldquoHyper Zagreb index of bridge and chain graphsrdquoOpen Journal of Mathematical Sciences vol 2 no 1 pp 1ndash172018

[21] A T Balaban I Motoc D Bonchev and O MekenyanldquoTopological indices for structureactivity correlationsrdquo inSteric Effects in Drug Design pp 21ndash55 Springer BerlinGermany 1983

[22] I Gutman B Ruscic N Trinajstic and C F Wilcox JrldquoGraph theory and molecular orbitals XII Acyclic polyenesrdquoe Journal of Chemical Physics vol 62 no 9 pp 3399ndash34051975

[23] W Gao M Asif and W Nazeer ldquoe study of honey combderived network via topological indicesrdquo Open Journal ofMathematical Analysis vol 2 no 2 pp 10ndash26 2018

[24] H Siddiqui and M R Farahani ldquoForgotten polynomial andforgotten index of certain interconnection networksrdquo OpenJournal of Mathematical Analysis vol 1 no 1 pp 44ndash592017

[25] W Gao B Muzaffar and W Nazeer ldquoK-Banhatti and K-hyper Banhatti indices of dominating David derived net-workrdquo Open Journal of Mathematical Analysis vol 1 no 1pp 13ndash24 2017

[26] W Gao A Asghar andW Nazeer ldquoComputing degree-basedtopological indices of jahangir graphrdquo Engineering and Ap-plied Science Letters vol 1 no 1 pp 16ndash22 2018

[27] T Doslic B Furtula A Graovac I Gutman S Moradi andZ Yarahmadi ldquoOn vertex degree based molecular structuredescriptorsrdquo MATCH Communications in Mathematical andin Computer Chemistry vol 66 no 2 pp 613ndash626 2011

[28] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975

[29] M Randic ldquoOn history of the Randic index and emerginghostility toward chemical graph theoryrdquo MATCH Commu-nications in Mathematical and in Computer Chemistryvol 59 pp 5ndash124 2008

[30] M Randic ldquoe connectivity index 25 years afterrdquo Journal ofMolecular Graphics and Modelling vol 20 no 1 pp 19ndash352001

[31] K C Das and I Gutman ldquoSome properties of the secondZagreb indexrdquo MATCH Communications in Mathematicaland in Computer Chemistry vol 52 no 1 pp 3ndash1 2004

[32] L B Kier and L H HallMolecular Connectivity in Chemistryand Drug Research Academic Press New York NY USA1976

[33] L B Kier and L H HallMolecular Connectivity in Structure-Activity Analysis Research Studies Boston MA USA 1986

[34] X Li and I Gutman Mathematical Aspects of Randic-TypeMolecular Structure Descriptors University of KragujevacKragujevac Serbia 2006

[35] B Bollobas and P Erdos ldquoGraphs of extremal weightsrdquo ArsCombinatoria vol 50 pp 225ndash233 1998

[36] Y Hu X Li Y Shi T Xu and I Gutman ldquoOn moleculargraphs with smallest and greatest zeroth-order general Randicindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 54 no 2 pp 425ndash434 2005

[37] Y Ma S Cao Y Shi I Gutman M Dehmer and B FurtulaldquoFrom the connectivity index to various Randic-type de-scriptorsrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 80 no 1 pp 85ndash106 2018

[38] I Gutman B Furtula and S B Bozkurt ldquoOn Randic energyrdquoLinear Algebra and its Applications vol 442 pp 50ndash57 2014

[39] X Li and Y Shi ldquoA survey on the Randic indexrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 59 no 1 pp 127ndash156 2008

[40] M Cavers S Fallat and S Kirkland ldquoOn the normalizedLaplacian energy and general Randic index R-1 of graphsrdquoLinear Algebra and its Applications vol 433 no 1 pp 172ndash190 2010

[41] S B Bozkurt A D Gungor I Gutman and A S CevikldquoRandic matrix and Randic energyrdquo MATCH Communica-tions in Mathematical and in Computer Chemistry vol 64pp 239ndash250 2010

[42] L Pogliani ldquoFrom molecular connectivity indices to semi-empirical connectivity terms recent trends in graph theo-retical descriptorsrdquo Chemical Reviews vol 100 no 10pp 3827ndash3858 2000

[43] J B Liu M Younas M Habib M Yousaf and W NazeerldquoM-polynomials and degree-based topological indices ofVC5C7[p q] and HC5C7[p q] nanotubesrdquo IEEE Accessvol 7 pp 41125ndash41132 2019

[44] M Riaz W Gao and A Q Baig ldquoM-polynomials and degree-based topological indices of some families of convex poly-topesrdquo Open Journal of Mathematical Sciences vol 2 no 1pp 18ndash28 2018

[45] M S Anjum and M U Safdar ldquoK Banhatti and K hyper-Banhatti indices of nanotubesrdquo Engineering and AppliedScience Letters vol 2 no 1 pp 19ndash37 2019

[46] Z Shao A R Virk M S Javed M A Rehman andM R Farahani ldquoDegree based graph invariants for themolecular graph of Bismuth Tri-Iodiderdquo Engineering andApplied Science Letters vol 2 no 1 pp 01ndash11 2019

[47] L Yan M R Farahani and W Gao ldquoDistance-based indicescomputation of symmetry molecular structuresrdquo OpenJournal of Mathematical Sciences vol 2 no 1 pp 323ndash3372018

[48] R Kanabur and S K Hosamani ldquoSome numerical invariantsassociated with V-phenylenic nanotube and nanotorirdquo En-gineering and Applied Science Letters vol 1 no 1 pp 1ndash92018

[49] J B Liu C Wang and S Wang ldquoZagreb indices and mul-tiplicative Zagreb indices of eulerian graphsrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 42 pp 67ndash782019

[50] J B Liu and X F Pan ldquoAsymptotic Laplacian-energy-likeinvariant of latticesrdquo Applied Mathematics and Computationvol 253 pp 205ndash214 2015

[51] J B Liu and X-F Pan ldquoMinimizing Kirchhoffindex amonggraphs with a given vertex bipartitenessrdquo Applied Mathe-matics and Computation vol 291 pp 84ndash88 2016

[52] J B Liu J Zhao and Z Zhu ldquoOn the number of spanningtrees and normalized Laplacian of linear octagonal-quadri-lateral networksrdquo International Journal of Quantum Chem-istry vol 2019 Article ID e25971 2019

Journal of Chemistry 11

TribologyAdvances in

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International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

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Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2018

Bioinorganic Chemistry and ApplicationsHindawiwwwhindawicom Volume 2018

SpectroscopyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Volume 2018

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SpectroscopyAnalytical ChemistryInternational Journal of

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Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

many papers and books on this topological invariant arepresent in the literature few of them are [31ndash34] Re-searchers recognized the importance of the Randic index indrug design Bollbas and Erdos famous mathematicians ofthat time investigated some hidden mathematical propertiesof RI [35] after that RI was worth studying and a surge ofpublications began [36ndash39] An unexpected mathematicalquality of the Randic index was discovered recently whichtells us about the relation of this topological invariant withthe normalized Laplacian matrix [38 40 41] e GRIknown as general RI [42] is defined as

Rα(G) 1113944uvisinE(G)

dudv( 1113857α (5)

23 M-Polynomial e mathematical formula of M-poly-nomial is

M(G x y) 1113944uvisinE(G)

xdu y

dv1113872 1113873 (6)

Detailed survey about the definitions of other TIscomputed in this paper and relation of TIs with M-poly-nomial can be seen in [43 44] where

Dx(f(x y)) xz

zx(f(x y))

Dy(f(x y)) yz

zy(f(x y))

Sx(f(x y)) 1113946x

0

f(t y)

tdt

Sy(f(x y)) 1113946y

0

f(x t)

tdt

J(f(x y)) J(f(x x))

Qα(f(x y)) xα(f(x y))

(7)

3 Methodology

To compute the M-polynomial of a graph G we need tocompute the number of vertices and edges in it and dividethe edge set into different classes with respect to the degreesof end vertices From the M-polynomials we can recovermany degree-dependent indices by applying some differ-ential and integral operators

4 Applications in Chemistry

A topological description is used to depict the features of thestudied compounds and indices of graph-theoretical originare used to investigate the correlations between structure andbiological activity [45ndash48] For example the Randic indexdemonstrates great relationship with the physical property ofalkanes e geometric arithmetic index has a similar role asthat of the Randic index e sum-connectivity index is

helpful in guessing the melting point of compounds Zagrebindices are used to calculate π-electron energy

5 Main Results

In this section we compute M-polynomials of understudynetworks and recover nine TIs from these polynomials

51 Results for OPn

Theorem 1 Let OPn be the swapped network then

M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(8)

Proof e OPn network has the following four types ofedges based on the degree of end vertices

E 13 OPn( 1113857 uv isin E OPn( 1113857 du 1 dv 31113864 1113865

E 22 OPn( 1113857 uv isin E OPn( 1113857 du 2 dv 21113864 1113865

E 23 OPn( 1113857 uv isin E OPn( 1113857 du 2 dv 31113864 1113865

E 33 OPn( 1113857 uv isin E OPn( 1113857 du 3 dv 31113864 1113865

(9)

such thatE 13 OPn( 1113857

11138681113868111386811138681113868111386811138681113868 2

E 22 OPn( 11138571113868111386811138681113868

1113868111386811138681113868 3

E 23 OPn( 11138571113868111386811138681113868

1113868111386811138681113868 (6n minus 14)

E 33 OPn( 11138571113868111386811138681113868

1113868111386811138681113868 3(n minus 2)(n minus 3)

21113888 1113889

(10)

Now from the definition of M-polynomial we have

M(G x y) 1113944δleilejleΔ

mijxiy

j

11139441le3

m13x1y3

+ 11139442le2

m22x2y2

+ 11139442le3

m23x2y3

+ 11139443le3

m33x3y3

E 13 OPn( 11138571113868111386811138681113868

1113868111386811138681113868x1y3

+ E 22 OPn( 11138571113868111386811138681113868

1113868111386811138681113868x2y2

+ E 23 OPn( 11138571113868111386811138681113868

1113868111386811138681113868x2y3

+ E 33 OPn( 11138571113868111386811138681113868

1113868111386811138681113868x3y3

2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(11)

Journal of Chemistry 3

Corollary 1 Let OPn be the swapped network then

M1 OPn( 1113857 9n2

minus 15n + 4 (12)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(13)

en

Dx(f(x y)) 2xy3

+ 6x2y2

+ 2(6n minus 14)x2y3

+9(n minus 2)(n minus 3)

21113888 1113889x

3y3

Dy(f(x y)) 6xy3

+ 6x2y2

+ 3(6n minus 14)x2y3

+9(n minus 2)(n minus 3)

21113888 1113889x

3y3

(14)

Now we have

M1 OPn( 1113857 Dxf +Dyf11138681113868111386811138681113868xy1

9n2

minus 15n + 4

(15)

Corollary 2 Let OPn be the swapped network then

M2 OPn( 1113857 27n2

2minus63n

2+ 15 (16)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(17)

en

Dy(f(x y)) 6xy3

+ 6x2y2

+ 3(6n minus 14)x2y3

+9(n minus 2)(n minus 3)

21113888 1113889x

3y3

DxDy(f(x y)) 6xy3

+ 12x2y2

+ 6(6n minus 14)x2y3

+27(n minus 2)(n minus 3)

21113888 1113889x

3y3

(18)

Now we have

M2 OPn( 1113857 DxDyf11138681113868111386811138681113868xy1

27n2

2minus63n

2+ 15

(19)

Corollary 3 Let OPn be the swapped network then

mM2 OPn( 1113857

n2

6+

n

6+

112

(20)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(21)

en

Sy(f(x y)) 23

1113874 1113875xy3

+32

1113874 1113875x2y2

+(6n minus 14)

31113888 1113889x

2y3

+(n minus 2)(n minus 3)

21113888 1113889x

3y3

SxSy(f(x y)) 23

1113874 1113875xy3

+34

1113874 1113875x2y2

+(6n minus 14)

61113888 1113889x

2y3

+(n minus 2)(n minus 3)

61113888 1113889x

3y3

(22)

Now we havem

M2 OPn( 1113857 SxSyf11138681113868111386811138681113868xy1

n2

6+

n

6+

112

(23)

Corollary 4 Let OPn be the swapped network then

Rα OPn( 1113857 6α + 6α2α + 3α2α(6n minus 14) +9α3α(n minus 2)(n minus 3)

2α1113888 1113889

(24)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(25)

4 Journal of Chemistry

en

Dαy(f(x y)) 6αxy

3+ 6αx

2y2

+ 3α(6n minus 14)x2y3

+9α(n minus 2)(n minus 3)

2α1113888 1113889x

3y3

DαxD

αy1113888(f(x y)) 6αxy

3+ 6α2αx

2y2

+ 3α2α(6n minus 14)x2y3

+9α3α(n minus 2)(n minus 3)

2α1113888 1113889x

3y31113889

(26)

Now we have

Rα OPn( 1113857 DαxD

αyf|xy11113872

6α + 6α2α + 3α2α(6n minus 14) +9α3α(n minus 2)(n minus 3)

2α

(27)

Corollary 5 Let OPn be the swapped network then

RRα OPn( 1113857 2α

3α+3α

22α+6n minus 143α2α

+(n minus 2)(n minus 3)

2α3α1113888 1113889

(28)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(29)

en

Sαy(f(x y))

2α

3α1113888 1113889xy

3+

3α

2α1113888 1113889x

2y2

+(6n minus 14)

3α1113888 1113889x

2y3

+(n minus 2)(n minus 3)

2α1113888 1113889x

3y3

SαxS

αy1113888(f(x y))

2α

3α1113888 1113889xy

3+

3α

22α1113888 1113889x

2y2

+(6n minus 14)

3α2α1113888 1113889x

2y3

+(n minus 2)(n minus 3)

2α3α1113888 1113889x

3y31113889

(30)

Now we have

RRα OPn( 1113857 SαxS

αyf|xy11113872

2α

3α+3α

22α+6n minus 143α2α

+(n minus 2)(n minus 3)

2α3α

(31)

Corollary 6 Let OPn be the swapped network then

SSD OPn( 1113857 3n2

minus 2n +13 (32)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(33)

en

SxDy(f(x y)) 6xy3

+ 3x2y2

+3(6n minus 14)

21113888 1113889x

2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

DxSy(f(x y)) 23

1113874 1113875xy3

+ 3x2y2

+2(6n minus 14)

31113888 1113889x

2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(34)

Now we have

SSD OPn( 1113857 DxSyf + SxDyf1113872 111387311138681113868111386811138681113868xy1

3n2

minus 2n +13

(35)

Corollary 7 Let OPn be the swapped network then

H OPn( 1113857 2n2

4minus

n

20minus

120

1113888 1113889 (36)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(37)

en

J(f(x y)) 5x4

+(6n minus 14)x5

+3(n minus 2)(n minus 3)

21113888 1113889x

6

2SxJ(f(x y)) 25x4

4+

(6n minus 14)x5

5+

(n minus 2)(n minus 3)x6

41113888 1113889

(38)

Journal of Chemistry 5

Now we have

H OPn( 1113857 2SxJf1113868111386811138681113868x1

2n2

4minus

n

20minus

120

1113888 1113889

(39)

Corollary 8 Let OPn be the swapped network then

I OPn( 1113857 9n2

4minus81n

20minus1295

(40)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(41)

en

Dy(f(x y)) 6xy3

+ 6x2y2

+ 3(6n minus 14)x2y3

+9(n minus 2)(n minus 3)

21113888 1113889x

3y3

DxDy(f(x y)) 6xy3

+ 12x2y2

+ 6(6n minus 14)x2y3

+27(n minus 2)(n minus 3)

21113888 1113889x

3y3

JDxDy(f(x y)) 18x4

+ 6(6n minus 14)x5

+27(n minus 2)(n minus 3)

21113888 1113889x

6

SxJDxDy(f(x y)) 18x4

4+

6(6n minus 14)

51113888 1113889x

5

+27(n minus 2)(n minus 3)

121113888 1113889x

6

(42)

Now we have

I OPn( 1113857 SxJDxDyf11138681113868111386811138681113868x1

9n2

4minus81n

20minus1295

(43)

Corollary 9 Let OPn be the swapped network then

A OPn( 1113857 729n2

64minus573n

64minus41332

(44)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(45)

enD

3y(f(x y)) 54xy

3+ 24x

2y2

+ 27(6n minus 14)x2y3

+81(n minus 2)(n minus 3)

21113888 1113889x

3y3

D3xD

3y(f(x y)) 54xy

3+ 192x

2y2

+ 216(6n minus 14)x2y3

+2187(n minus 2)(n minus 3)

21113888 1113889x

3y3

JD3xD

3y(f(x y)) 246x

4+ 216(6n minus 14)x

5

+2187(n minus 2)(n minus 3)

21113888 1113889x

6

Qminus 2JD3xD

3y(f(x y)) 246x

2+ 216(6n minus 14)x

3

+2187(n minus 2)(n minus 3)

21113888 1113889x

4

S3xQminus 2JD

3xD

3y(f(x y))

123x2

4+ 8(6n minus 14)x

3

+729(n minus 2)(n minus 3)

641113888 1113889x

4

(46)Now we have

A OPn( 1113857 S3xQminus 2JD

3xD

3yf

11138681113868111386811138681113868x1

729n2

64minus573n

64minus41332

(47)

52 Results for (ORk)

Theorem 2 Let ORk be the swapped network then

M(G x y) nkxky

k+1+

n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(48)

Proof e ORk network has the following two types ofedges based on the degree of end vertices

E kk+1 ORk( 1113857 uv isin E ORk( 1113857 du k dv k + 11113864 1113865

E k+1k+1 ORk( 1113857 uv isin E ORk( 1113857 du k + 1 dv k + 11113864 1113865

(49)

6 Journal of Chemistry

such that

E kk+1 ORk( 11138571113868111386811138681113868

1113868111386811138681113868 nk

E k+1k+1 ORk( 11138571113868111386811138681113868

1113868111386811138681113868 n2(k + 1) minus n(1 + 2k)

2

(50)

Now from the definition of M-polynomial we have

M(G x y) 1113944δleilejleΔ

mijxiy

j

1113944klek+1

mk(k+1)xky

k+1+ 1113944

k+1lek+1m(k+1)(k+1)x

k+1y

k+1

E kk+1 ORk( 11138571113868111386811138681113868

1113868111386811138681113868xky

k+1+ E k+1k+1 ORk( 1113857

11138681113868111386811138681113868111386811138681113868x

k+1y

k+1

nkxky

k+1+

n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(51)

Corollary 10 Let (ORk) be the swapped network then

M1 ORk( 1113857 2nk2

+ nk + n2(k + 1)

2minus n(k + 1)(1 + 2k)

(52)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(53)

en

Dx(f(x y)) nk2x

ky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

Dy(f(x y)) nk(k + 1)xky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(54)

Now we have

M1 ORk( 1113857 Dxf + Dyf11138681113868111386811138681113868xy1

2nk2

+ nk + n2(k + 1)

2minus n(k + 1)(1 + 2k)

(55)

Corollary 11 Let (ORk) be the swapped network then

M2 ORk( 1113857 nk3

+ nk2

+n2(k + 1)3

2minus

n(k + 1)2(1 + 2k)

2

(56)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(57)

en

Dy(f(x y)) nk2x

ky

k+1+ nkx

ky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxDy(f(x y)) nk3x

ky

k+1+ nk

2x

ky

k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(58)

Now we have

M2 ORk( 1113857 DxDyf11138681113868111386811138681113868xy1

nk3

+ nk2

+n2(k + 1)3

2minus

n(k + 1)2(1 + 2k)

2

(59)

Corollary 12 Let (ORk) be the swapped network then

mM2 ORk( 1113857

nk

k2 + k( )+

n2

(2k + 2)minus

n(1 + 2k)

2(k + 1)2 (60)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(61)

en

Sy(f(x y)) nk

(k + 1)1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2(k + 1)1113888 1113889x

k+1y

k+1

SxSy(f(x y)) nk

k2 + k( )1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2(k + 1)21113888 1113889x

k+1y

k+1

(62)

Journal of Chemistry 7

Now we havem

M2 ORk( 1113857 SxSyf11138681113868111386811138681113868xy1

nk

k2 + k( )+

n2

(2k + 2)minus

n(1 + 2k)

2(k + 1)2

(63)

Corollary 13 Let (ORk) be the swapped network then

Rα1113888 ORk( 1113857 nk3α

+ nk2α

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

2α1113888 1113889x

k+1y

k+11113889

(64)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(65)

en

Dαy(f(x y)) nk

2αx

ky

k+1+ nk

αx

ky

k+1

+(k + 1)α n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DαxD

αy1113888(f(x y)) nk

3αx

ky

k+1+ nk

2αx

ky

k+1

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+11113889

(66)

Now we have

Rα ORk( 1113857 DαxD

αyf

11138681113868111386811138681113868xy11113874

nk3α

+ nk2α

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

2α1113888 1113889x

k+1y

k+1

(67)

Corollary 14 Let (ORk) be the swapped network then

RRα ORk( 1113857 n

(k + 1)α+

n2(k + 1) minus n(1 + 2k)

(k + 1)2α2α1113888 11138891113888 1113889

(68)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(69)

en

Sαy(f(x y))

nkα

(k + 1)α1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2α(k + 1)α1113888 1113889x

k+1y

k+1

SαxS

αy1113888(f(x y))

n

(k + 1)α1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2α(k + 1)2α1113888 1113889x

k+1y

k+11113889

(70)

Now we have

RRα ORk( 1113857 SαxS

αyf

11138681113868111386811138681113868xy11113874

n

(k + 1)α+

n2(k + 1) minus n(1 + 2k)

(k + 1)2α2α1113888 1113889

(71)

Corollary 15 Let (ORk) be the swapped network then

SSD ORk( 1113857 nk2

(k + 1)+ nk + n + n

2(k + 1) minus n(1 + 2k)

(72)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(73)

en

SxDy(f(x y)) nkxky

k+1+ nx

ky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxSy(f(x y)) nk2

(k + 1)1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(74)

Now we have

SSD ORk( 1113857 DxSyf + SxDyf1113872 111387311138681113868111386811138681113868xy1

nk2

(k + 1)+ nk + n + n

2(k + 1) minus n(1 + 2k)

(75)

8 Journal of Chemistry

Corollary 16 Let (ORk) be the swapped network then

H ORk( 1113857 2nk

2k + 1+

n2(k + 1)

4k + 4minus

n(1 + 2k)

4k + 41113888 1113889 (76)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(77)

en

J(f(x y)) nkx2k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

2SxJ(f(x y)) 21113888nk

2k + 11113888 1113889x

2k+1

+n2(k + 1)

4k + 41113888 1113889x

2k+2

minusn(1 + 2k)

4k + 41113888 1113889x

2k+21113889

(78)

Now we have

H ORk( 1113857 2SxJf1113868111386811138681113868x1

2nk

2k + 1+

n2(k + 1)

4k + 4minus

n(1 + 2k)

4k + 41113888 1113889

(79)

Corollary 17 Let (ORk) be the swapped network then

I ORk( 1113857 nk3

2k + 1+

nk2

2k + 1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889

(80)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(81)

en

Dy(f(x y)) nk(k + 1)xky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxDy(f(x y)) nk3x

ky

k+1+ nk

2x

ky

k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

JDxDy(f(x y)) nk3x2k+1

+ nk2x2k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

SxJDxDy(f(x y)) nk3

2k + 11113888 1113889x

2k+1+

nk2

2k + 11113888 1113889x

2k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889x

2k+2

(82)

Now we have

I ORk( 1113857 SxJDxDyf11138681113868111386811138681113868x1

nk3

2k + 1+

nk2

2k + 1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889

(83)

Corollary 18 Let (ORk) be the swapped network then

A ORk( 1113857 nk5(k + 1)2

(2k minus 1)3

+nk4(k + 1)2

(2k minus 1)3

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889

(84)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(85)

Journal of Chemistry 9

en

D3y(f(x y)) nk

2(k + 1)

2x

ky

k+1+ nk(k + 1)

2x

ky

k+1

+(k + 1)3 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

D3yD

3y(f(x y)) nk

5(k + 1)

2x

ky

k+1+ nk

4(k + 1)

2x

ky

k+1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

JD3yD

3y(f(x y)) nk

5(k + 1)

2x2k+1

+ nk4(k + 1)

2x2k+1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

Qminus 2JD3yD

3y(f(x y)) nk

5(k + 1)

2x2kminus 1

+ nk4(k + 1)

2x2kminus 1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k

S3xQminus 2JD

3yD

3y(f(x y))

nk5(k + 1)2

(2k minus 1)31113888 1113889x

2kminus 1+

nk4(k + 1)2

(2k minus 1)31113888 1113889x

2kminus 1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889x2k

(86)

Now we have

A ORk( 1113857 S3xQminus 2JD

3yD

3yf

11138681113868111386811138681113868x1

nk5(k + 1)2

(2k minus 1)3+

nk4(k + 1)2

(2k minus 1)3

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889

(87)

6 Conclusion

In this paper our focus is on swapped interconnectionnetworks that allow systematic construction of large scal-able modular and robust parallel architectures whilemaintaining many desirable attributes of the underlyingbasis network comprising its clusters We have computedseveral TIs of underlined networks Firstly we computed M-polynomials of understudy networks and then we recoveredZagreb indices Randic indices and some other indices[49ndash52]

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to this study

Acknowledgments

is study was partially supported by the Natural ScienceFoundation of Anhui Province (no KJ2018A059)

References

[1] W Gao M R Farahani and L Shi ldquoForgotten topologicalindex of some drug structuresrdquo Acta Medica Mediterraneavol 32 no 1 pp 579ndash585 2016

[2] W Gao M K Siddiqui M Imran M K Jamil andM R Farahani ldquoForgotten topological index of chemicalstructure in drugsrdquo Saudi Pharmaceutical Journal vol 24no 3 pp 258ndash264 2016

[3] W Gao W Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 Article ID 3216327 8 pages 2016

[4] M Imran A Asghar and A Q Baig ldquoOn graph invariants ofoxide networkrdquo Engineering and Applied Science Lettersvol 1(2018) no 1 pp 23ndash28 2018

[5] G Liu Z Jia and W Gao ldquoOntology similarity computingbased on stochastic primal dual coordinate techniquerdquo OpenJournal of Mathematical Sciences vol 2(2018) no 1pp 221ndash227 2018

[6] S Noreen and A Mahmood ldquoZagreb polynomials andredefined Zagreb indices for the line graph of carbonnanoconesrdquo Open Journal of Mathematical Analysisvol 2(2018) no 1 pp 66ndash73 2018

[7] S Hayat and M Imran ldquoComputation of topological indicesof certain networksrdquo Applied Mathematics and Computationvol 240 pp 213ndash228 2014

[8] S Hayat and M Imran ldquoComputation of certain topologicalindices of nanotubes covered byrdquo Journal of Computationaland eoretical Nanoscience vol 12 no 4 pp 533ndash541 2015

[9] A Q Baig M Imran and H Ali ldquoOn topological indices ofpoly oxide poly silicate DOX and DSL networksrdquo CanadianJournal of Chemistry vol 93 no 7 pp 730ndash739 2015

[10] Y Bashir A Aslam M Kamran et al ldquoOn forgotten topo-logical indices of some dendrimers structurerdquo Moleculesvol 22 no 6 p 867 2017

[11] A Aslam J L Guirao S Ahmad and W Gao ldquoTopologicalindices of the line graph of subdivision graph of completebipartite graphsrdquo Applied Mathematics amp Information Sci-ences vol 11 no 6 pp 1631ndash1636 2017

[12] G C Marsden P J Marchand P Harvey and S C EsenerldquoOptical transpose interconnection system architecturesrdquoOptics Letters vol 18 no 13 pp 1083ndash1085 1993

[13] A Al-Ayyoub A Awwad K Day and M Ould-KhaoualdquoGeneralized methods for algorithm development on opticalsystemsrdquo e Journal of Supercomputing vol 38 no 2pp 111ndash125 2006

[14] C F Wang and S Sahni ldquoMatrix multiplication on the OTIS-mesh optoelectronic computerrdquo IEEE Transactions onComputers vol 50 no 7 pp 635ndash646 2001

[15] C F Wang and S Sahni ldquoImage processing on the OTIS-mesh optoelectronic computerrdquo IEEE Transactions on Paralleland Distributed Systems vol 11 no 2 pp 97ndash109 2000

[16] C F Wang and S Sahni ldquoBasic operations on the OTIS-meshoptoelectronic computerrdquo IEEE Transactions on Parallel andDistributed Systems vol 9 no 12 pp 1226ndash1236 1998

10 Journal of Chemistry

[17] S Rajasekaran and S Sahni ldquoRandomized routing selectionand sorting on the OTIS-meshrdquo IEEE Transactions on Paralleland Distributed Systems vol 9 no 9 pp 833ndash840 1998

[18] A R Virk M N Jhangeer and M A Rehman ldquoReverseZagreb and reverse hyper-zagreb indices for silicon carbideSi2C3I[rs] and Si2C3II[rs]rdquo Engineering and Applied ScienceLetters vol 1 no 2 pp 37ndash50 2018

[19] N De ldquoComputing reformulated first Zagreb index of somechemical graphs as an application of generalized hierarchicalproduct of graphsrdquo Open Journal of Mathematical Sciencesvol 2 no 1 pp 338ndash350 2018

[20] N De ldquoHyper Zagreb index of bridge and chain graphsrdquoOpen Journal of Mathematical Sciences vol 2 no 1 pp 1ndash172018

[21] A T Balaban I Motoc D Bonchev and O MekenyanldquoTopological indices for structureactivity correlationsrdquo inSteric Effects in Drug Design pp 21ndash55 Springer BerlinGermany 1983

[22] I Gutman B Ruscic N Trinajstic and C F Wilcox JrldquoGraph theory and molecular orbitals XII Acyclic polyenesrdquoe Journal of Chemical Physics vol 62 no 9 pp 3399ndash34051975

[23] W Gao M Asif and W Nazeer ldquoe study of honey combderived network via topological indicesrdquo Open Journal ofMathematical Analysis vol 2 no 2 pp 10ndash26 2018

[24] H Siddiqui and M R Farahani ldquoForgotten polynomial andforgotten index of certain interconnection networksrdquo OpenJournal of Mathematical Analysis vol 1 no 1 pp 44ndash592017

[25] W Gao B Muzaffar and W Nazeer ldquoK-Banhatti and K-hyper Banhatti indices of dominating David derived net-workrdquo Open Journal of Mathematical Analysis vol 1 no 1pp 13ndash24 2017

[26] W Gao A Asghar andW Nazeer ldquoComputing degree-basedtopological indices of jahangir graphrdquo Engineering and Ap-plied Science Letters vol 1 no 1 pp 16ndash22 2018

[27] T Doslic B Furtula A Graovac I Gutman S Moradi andZ Yarahmadi ldquoOn vertex degree based molecular structuredescriptorsrdquo MATCH Communications in Mathematical andin Computer Chemistry vol 66 no 2 pp 613ndash626 2011

[28] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975

[29] M Randic ldquoOn history of the Randic index and emerginghostility toward chemical graph theoryrdquo MATCH Commu-nications in Mathematical and in Computer Chemistryvol 59 pp 5ndash124 2008

[30] M Randic ldquoe connectivity index 25 years afterrdquo Journal ofMolecular Graphics and Modelling vol 20 no 1 pp 19ndash352001

[31] K C Das and I Gutman ldquoSome properties of the secondZagreb indexrdquo MATCH Communications in Mathematicaland in Computer Chemistry vol 52 no 1 pp 3ndash1 2004

[32] L B Kier and L H HallMolecular Connectivity in Chemistryand Drug Research Academic Press New York NY USA1976

[33] L B Kier and L H HallMolecular Connectivity in Structure-Activity Analysis Research Studies Boston MA USA 1986

[34] X Li and I Gutman Mathematical Aspects of Randic-TypeMolecular Structure Descriptors University of KragujevacKragujevac Serbia 2006

[35] B Bollobas and P Erdos ldquoGraphs of extremal weightsrdquo ArsCombinatoria vol 50 pp 225ndash233 1998

[36] Y Hu X Li Y Shi T Xu and I Gutman ldquoOn moleculargraphs with smallest and greatest zeroth-order general Randicindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 54 no 2 pp 425ndash434 2005

[37] Y Ma S Cao Y Shi I Gutman M Dehmer and B FurtulaldquoFrom the connectivity index to various Randic-type de-scriptorsrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 80 no 1 pp 85ndash106 2018

[38] I Gutman B Furtula and S B Bozkurt ldquoOn Randic energyrdquoLinear Algebra and its Applications vol 442 pp 50ndash57 2014

[39] X Li and Y Shi ldquoA survey on the Randic indexrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 59 no 1 pp 127ndash156 2008

[40] M Cavers S Fallat and S Kirkland ldquoOn the normalizedLaplacian energy and general Randic index R-1 of graphsrdquoLinear Algebra and its Applications vol 433 no 1 pp 172ndash190 2010

[41] S B Bozkurt A D Gungor I Gutman and A S CevikldquoRandic matrix and Randic energyrdquo MATCH Communica-tions in Mathematical and in Computer Chemistry vol 64pp 239ndash250 2010

[42] L Pogliani ldquoFrom molecular connectivity indices to semi-empirical connectivity terms recent trends in graph theo-retical descriptorsrdquo Chemical Reviews vol 100 no 10pp 3827ndash3858 2000

[43] J B Liu M Younas M Habib M Yousaf and W NazeerldquoM-polynomials and degree-based topological indices ofVC5C7[p q] and HC5C7[p q] nanotubesrdquo IEEE Accessvol 7 pp 41125ndash41132 2019

[44] M Riaz W Gao and A Q Baig ldquoM-polynomials and degree-based topological indices of some families of convex poly-topesrdquo Open Journal of Mathematical Sciences vol 2 no 1pp 18ndash28 2018

[45] M S Anjum and M U Safdar ldquoK Banhatti and K hyper-Banhatti indices of nanotubesrdquo Engineering and AppliedScience Letters vol 2 no 1 pp 19ndash37 2019

[46] Z Shao A R Virk M S Javed M A Rehman andM R Farahani ldquoDegree based graph invariants for themolecular graph of Bismuth Tri-Iodiderdquo Engineering andApplied Science Letters vol 2 no 1 pp 01ndash11 2019

[47] L Yan M R Farahani and W Gao ldquoDistance-based indicescomputation of symmetry molecular structuresrdquo OpenJournal of Mathematical Sciences vol 2 no 1 pp 323ndash3372018

[48] R Kanabur and S K Hosamani ldquoSome numerical invariantsassociated with V-phenylenic nanotube and nanotorirdquo En-gineering and Applied Science Letters vol 1 no 1 pp 1ndash92018

[49] J B Liu C Wang and S Wang ldquoZagreb indices and mul-tiplicative Zagreb indices of eulerian graphsrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 42 pp 67ndash782019

[50] J B Liu and X F Pan ldquoAsymptotic Laplacian-energy-likeinvariant of latticesrdquo Applied Mathematics and Computationvol 253 pp 205ndash214 2015

[51] J B Liu and X-F Pan ldquoMinimizing Kirchhoffindex amonggraphs with a given vertex bipartitenessrdquo Applied Mathe-matics and Computation vol 291 pp 84ndash88 2016

[52] J B Liu J Zhao and Z Zhu ldquoOn the number of spanningtrees and normalized Laplacian of linear octagonal-quadri-lateral networksrdquo International Journal of Quantum Chem-istry vol 2019 Article ID e25971 2019

Journal of Chemistry 11

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Na

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ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

Corollary 1 Let OPn be the swapped network then

M1 OPn( 1113857 9n2

minus 15n + 4 (12)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(13)

en

Dx(f(x y)) 2xy3

+ 6x2y2

+ 2(6n minus 14)x2y3

+9(n minus 2)(n minus 3)

21113888 1113889x

3y3

Dy(f(x y)) 6xy3

+ 6x2y2

+ 3(6n minus 14)x2y3

+9(n minus 2)(n minus 3)

21113888 1113889x

3y3

(14)

Now we have

M1 OPn( 1113857 Dxf +Dyf11138681113868111386811138681113868xy1

9n2

minus 15n + 4

(15)

Corollary 2 Let OPn be the swapped network then

M2 OPn( 1113857 27n2

2minus63n

2+ 15 (16)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(17)

en

Dy(f(x y)) 6xy3

+ 6x2y2

+ 3(6n minus 14)x2y3

+9(n minus 2)(n minus 3)

21113888 1113889x

3y3

DxDy(f(x y)) 6xy3

+ 12x2y2

+ 6(6n minus 14)x2y3

+27(n minus 2)(n minus 3)

21113888 1113889x

3y3

(18)

Now we have

M2 OPn( 1113857 DxDyf11138681113868111386811138681113868xy1

27n2

2minus63n

2+ 15

(19)

Corollary 3 Let OPn be the swapped network then

mM2 OPn( 1113857

n2

6+

n

6+

112

(20)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(21)

en

Sy(f(x y)) 23

1113874 1113875xy3

+32

1113874 1113875x2y2

+(6n minus 14)

31113888 1113889x

2y3

+(n minus 2)(n minus 3)

21113888 1113889x

3y3

SxSy(f(x y)) 23

1113874 1113875xy3

+34

1113874 1113875x2y2

+(6n minus 14)

61113888 1113889x

2y3

+(n minus 2)(n minus 3)

61113888 1113889x

3y3

(22)

Now we havem

M2 OPn( 1113857 SxSyf11138681113868111386811138681113868xy1

n2

6+

n

6+

112

(23)

Corollary 4 Let OPn be the swapped network then

Rα OPn( 1113857 6α + 6α2α + 3α2α(6n minus 14) +9α3α(n minus 2)(n minus 3)

2α1113888 1113889

(24)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(25)

4 Journal of Chemistry

en

Dαy(f(x y)) 6αxy

3+ 6αx

2y2

+ 3α(6n minus 14)x2y3

+9α(n minus 2)(n minus 3)

2α1113888 1113889x

3y3

DαxD

αy1113888(f(x y)) 6αxy

3+ 6α2αx

2y2

+ 3α2α(6n minus 14)x2y3

+9α3α(n minus 2)(n minus 3)

2α1113888 1113889x

3y31113889

(26)

Now we have

Rα OPn( 1113857 DαxD

αyf|xy11113872

6α + 6α2α + 3α2α(6n minus 14) +9α3α(n minus 2)(n minus 3)

2α

(27)

Corollary 5 Let OPn be the swapped network then

RRα OPn( 1113857 2α

3α+3α

22α+6n minus 143α2α

+(n minus 2)(n minus 3)

2α3α1113888 1113889

(28)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(29)

en

Sαy(f(x y))

2α

3α1113888 1113889xy

3+

3α

2α1113888 1113889x

2y2

+(6n minus 14)

3α1113888 1113889x

2y3

+(n minus 2)(n minus 3)

2α1113888 1113889x

3y3

SαxS

αy1113888(f(x y))

2α

3α1113888 1113889xy

3+

3α

22α1113888 1113889x

2y2

+(6n minus 14)

3α2α1113888 1113889x

2y3

+(n minus 2)(n minus 3)

2α3α1113888 1113889x

3y31113889

(30)

Now we have

RRα OPn( 1113857 SαxS

αyf|xy11113872

2α

3α+3α

22α+6n minus 143α2α

+(n minus 2)(n minus 3)

2α3α

(31)

Corollary 6 Let OPn be the swapped network then

SSD OPn( 1113857 3n2

minus 2n +13 (32)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(33)

en

SxDy(f(x y)) 6xy3

+ 3x2y2

+3(6n minus 14)

21113888 1113889x

2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

DxSy(f(x y)) 23

1113874 1113875xy3

+ 3x2y2

+2(6n minus 14)

31113888 1113889x

2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(34)

Now we have

SSD OPn( 1113857 DxSyf + SxDyf1113872 111387311138681113868111386811138681113868xy1

3n2

minus 2n +13

(35)

Corollary 7 Let OPn be the swapped network then

H OPn( 1113857 2n2

4minus

n

20minus

120

1113888 1113889 (36)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(37)

en

J(f(x y)) 5x4

+(6n minus 14)x5

+3(n minus 2)(n minus 3)

21113888 1113889x

6

2SxJ(f(x y)) 25x4

4+

(6n minus 14)x5

5+

(n minus 2)(n minus 3)x6

41113888 1113889

(38)

Journal of Chemistry 5

Now we have

H OPn( 1113857 2SxJf1113868111386811138681113868x1

2n2

4minus

n

20minus

120

1113888 1113889

(39)

Corollary 8 Let OPn be the swapped network then

I OPn( 1113857 9n2

4minus81n

20minus1295

(40)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(41)

en

Dy(f(x y)) 6xy3

+ 6x2y2

+ 3(6n minus 14)x2y3

+9(n minus 2)(n minus 3)

21113888 1113889x

3y3

DxDy(f(x y)) 6xy3

+ 12x2y2

+ 6(6n minus 14)x2y3

+27(n minus 2)(n minus 3)

21113888 1113889x

3y3

JDxDy(f(x y)) 18x4

+ 6(6n minus 14)x5

+27(n minus 2)(n minus 3)

21113888 1113889x

6

SxJDxDy(f(x y)) 18x4

4+

6(6n minus 14)

51113888 1113889x

5

+27(n minus 2)(n minus 3)

121113888 1113889x

6

(42)

Now we have

I OPn( 1113857 SxJDxDyf11138681113868111386811138681113868x1

9n2

4minus81n

20minus1295

(43)

Corollary 9 Let OPn be the swapped network then

A OPn( 1113857 729n2

64minus573n

64minus41332

(44)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(45)

enD

3y(f(x y)) 54xy

3+ 24x

2y2

+ 27(6n minus 14)x2y3

+81(n minus 2)(n minus 3)

21113888 1113889x

3y3

D3xD

3y(f(x y)) 54xy

3+ 192x

2y2

+ 216(6n minus 14)x2y3

+2187(n minus 2)(n minus 3)

21113888 1113889x

3y3

JD3xD

3y(f(x y)) 246x

4+ 216(6n minus 14)x

5

+2187(n minus 2)(n minus 3)

21113888 1113889x

6

Qminus 2JD3xD

3y(f(x y)) 246x

2+ 216(6n minus 14)x

3

+2187(n minus 2)(n minus 3)

21113888 1113889x

4

S3xQminus 2JD

3xD

3y(f(x y))

123x2

4+ 8(6n minus 14)x

3

+729(n minus 2)(n minus 3)

641113888 1113889x

4

(46)Now we have

A OPn( 1113857 S3xQminus 2JD

3xD

3yf

11138681113868111386811138681113868x1

729n2

64minus573n

64minus41332

(47)

52 Results for (ORk)

Theorem 2 Let ORk be the swapped network then

M(G x y) nkxky

k+1+

n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(48)

Proof e ORk network has the following two types ofedges based on the degree of end vertices

E kk+1 ORk( 1113857 uv isin E ORk( 1113857 du k dv k + 11113864 1113865

E k+1k+1 ORk( 1113857 uv isin E ORk( 1113857 du k + 1 dv k + 11113864 1113865

(49)

6 Journal of Chemistry

such that

E kk+1 ORk( 11138571113868111386811138681113868

1113868111386811138681113868 nk

E k+1k+1 ORk( 11138571113868111386811138681113868

1113868111386811138681113868 n2(k + 1) minus n(1 + 2k)

2

(50)

Now from the definition of M-polynomial we have

M(G x y) 1113944δleilejleΔ

mijxiy

j

1113944klek+1

mk(k+1)xky

k+1+ 1113944

k+1lek+1m(k+1)(k+1)x

k+1y

k+1

E kk+1 ORk( 11138571113868111386811138681113868

1113868111386811138681113868xky

k+1+ E k+1k+1 ORk( 1113857

11138681113868111386811138681113868111386811138681113868x

k+1y

k+1

nkxky

k+1+

n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(51)

Corollary 10 Let (ORk) be the swapped network then

M1 ORk( 1113857 2nk2

+ nk + n2(k + 1)

2minus n(k + 1)(1 + 2k)

(52)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(53)

en

Dx(f(x y)) nk2x

ky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

Dy(f(x y)) nk(k + 1)xky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(54)

Now we have

M1 ORk( 1113857 Dxf + Dyf11138681113868111386811138681113868xy1

2nk2

+ nk + n2(k + 1)

2minus n(k + 1)(1 + 2k)

(55)

Corollary 11 Let (ORk) be the swapped network then

M2 ORk( 1113857 nk3

+ nk2

+n2(k + 1)3

2minus

n(k + 1)2(1 + 2k)

2

(56)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(57)

en

Dy(f(x y)) nk2x

ky

k+1+ nkx

ky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxDy(f(x y)) nk3x

ky

k+1+ nk

2x

ky

k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(58)

Now we have

M2 ORk( 1113857 DxDyf11138681113868111386811138681113868xy1

nk3

+ nk2

+n2(k + 1)3

2minus

n(k + 1)2(1 + 2k)

2

(59)

Corollary 12 Let (ORk) be the swapped network then

mM2 ORk( 1113857

nk

k2 + k( )+

n2

(2k + 2)minus

n(1 + 2k)

2(k + 1)2 (60)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(61)

en

Sy(f(x y)) nk

(k + 1)1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2(k + 1)1113888 1113889x

k+1y

k+1

SxSy(f(x y)) nk

k2 + k( )1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2(k + 1)21113888 1113889x

k+1y

k+1

(62)

Journal of Chemistry 7

Now we havem

M2 ORk( 1113857 SxSyf11138681113868111386811138681113868xy1

nk

k2 + k( )+

n2

(2k + 2)minus

n(1 + 2k)

2(k + 1)2

(63)

Corollary 13 Let (ORk) be the swapped network then

Rα1113888 ORk( 1113857 nk3α

+ nk2α

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

2α1113888 1113889x

k+1y

k+11113889

(64)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(65)

en

Dαy(f(x y)) nk

2αx

ky

k+1+ nk

αx

ky

k+1

+(k + 1)α n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DαxD

αy1113888(f(x y)) nk

3αx

ky

k+1+ nk

2αx

ky

k+1

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+11113889

(66)

Now we have

Rα ORk( 1113857 DαxD

αyf

11138681113868111386811138681113868xy11113874

nk3α

+ nk2α

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

2α1113888 1113889x

k+1y

k+1

(67)

Corollary 14 Let (ORk) be the swapped network then

RRα ORk( 1113857 n

(k + 1)α+

n2(k + 1) minus n(1 + 2k)

(k + 1)2α2α1113888 11138891113888 1113889

(68)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(69)

en

Sαy(f(x y))

nkα

(k + 1)α1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2α(k + 1)α1113888 1113889x

k+1y

k+1

SαxS

αy1113888(f(x y))

n

(k + 1)α1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2α(k + 1)2α1113888 1113889x

k+1y

k+11113889

(70)

Now we have

RRα ORk( 1113857 SαxS

αyf

11138681113868111386811138681113868xy11113874

n

(k + 1)α+

n2(k + 1) minus n(1 + 2k)

(k + 1)2α2α1113888 1113889

(71)

Corollary 15 Let (ORk) be the swapped network then

SSD ORk( 1113857 nk2

(k + 1)+ nk + n + n

2(k + 1) minus n(1 + 2k)

(72)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(73)

en

SxDy(f(x y)) nkxky

k+1+ nx

ky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxSy(f(x y)) nk2

(k + 1)1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(74)

Now we have

SSD ORk( 1113857 DxSyf + SxDyf1113872 111387311138681113868111386811138681113868xy1

nk2

(k + 1)+ nk + n + n

2(k + 1) minus n(1 + 2k)

(75)

8 Journal of Chemistry

Corollary 16 Let (ORk) be the swapped network then

H ORk( 1113857 2nk

2k + 1+

n2(k + 1)

4k + 4minus

n(1 + 2k)

4k + 41113888 1113889 (76)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(77)

en

J(f(x y)) nkx2k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

2SxJ(f(x y)) 21113888nk

2k + 11113888 1113889x

2k+1

+n2(k + 1)

4k + 41113888 1113889x

2k+2

minusn(1 + 2k)

4k + 41113888 1113889x

2k+21113889

(78)

Now we have

H ORk( 1113857 2SxJf1113868111386811138681113868x1

2nk

2k + 1+

n2(k + 1)

4k + 4minus

n(1 + 2k)

4k + 41113888 1113889

(79)

Corollary 17 Let (ORk) be the swapped network then

I ORk( 1113857 nk3

2k + 1+

nk2

2k + 1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889

(80)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(81)

en

Dy(f(x y)) nk(k + 1)xky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxDy(f(x y)) nk3x

ky

k+1+ nk

2x

ky

k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

JDxDy(f(x y)) nk3x2k+1

+ nk2x2k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

SxJDxDy(f(x y)) nk3

2k + 11113888 1113889x

2k+1+

nk2

2k + 11113888 1113889x

2k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889x

2k+2

(82)

Now we have

I ORk( 1113857 SxJDxDyf11138681113868111386811138681113868x1

nk3

2k + 1+

nk2

2k + 1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889

(83)

Corollary 18 Let (ORk) be the swapped network then

A ORk( 1113857 nk5(k + 1)2

(2k minus 1)3

+nk4(k + 1)2

(2k minus 1)3

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889

(84)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(85)

Journal of Chemistry 9

en

D3y(f(x y)) nk

2(k + 1)

2x

ky

k+1+ nk(k + 1)

2x

ky

k+1

+(k + 1)3 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

D3yD

3y(f(x y)) nk

5(k + 1)

2x

ky

k+1+ nk

4(k + 1)

2x

ky

k+1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

JD3yD

3y(f(x y)) nk

5(k + 1)

2x2k+1

+ nk4(k + 1)

2x2k+1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

Qminus 2JD3yD

3y(f(x y)) nk

5(k + 1)

2x2kminus 1

+ nk4(k + 1)

2x2kminus 1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k

S3xQminus 2JD

3yD

3y(f(x y))

nk5(k + 1)2

(2k minus 1)31113888 1113889x

2kminus 1+

nk4(k + 1)2

(2k minus 1)31113888 1113889x

2kminus 1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889x2k

(86)

Now we have

A ORk( 1113857 S3xQminus 2JD

3yD

3yf

11138681113868111386811138681113868x1

nk5(k + 1)2

(2k minus 1)3+

nk4(k + 1)2

(2k minus 1)3

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889

(87)

6 Conclusion

In this paper our focus is on swapped interconnectionnetworks that allow systematic construction of large scal-able modular and robust parallel architectures whilemaintaining many desirable attributes of the underlyingbasis network comprising its clusters We have computedseveral TIs of underlined networks Firstly we computed M-polynomials of understudy networks and then we recoveredZagreb indices Randic indices and some other indices[49ndash52]

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to this study

Acknowledgments

is study was partially supported by the Natural ScienceFoundation of Anhui Province (no KJ2018A059)

References

[1] W Gao M R Farahani and L Shi ldquoForgotten topologicalindex of some drug structuresrdquo Acta Medica Mediterraneavol 32 no 1 pp 579ndash585 2016

[2] W Gao M K Siddiqui M Imran M K Jamil andM R Farahani ldquoForgotten topological index of chemicalstructure in drugsrdquo Saudi Pharmaceutical Journal vol 24no 3 pp 258ndash264 2016

[3] W Gao W Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 Article ID 3216327 8 pages 2016

[4] M Imran A Asghar and A Q Baig ldquoOn graph invariants ofoxide networkrdquo Engineering and Applied Science Lettersvol 1(2018) no 1 pp 23ndash28 2018

[5] G Liu Z Jia and W Gao ldquoOntology similarity computingbased on stochastic primal dual coordinate techniquerdquo OpenJournal of Mathematical Sciences vol 2(2018) no 1pp 221ndash227 2018

[6] S Noreen and A Mahmood ldquoZagreb polynomials andredefined Zagreb indices for the line graph of carbonnanoconesrdquo Open Journal of Mathematical Analysisvol 2(2018) no 1 pp 66ndash73 2018

[7] S Hayat and M Imran ldquoComputation of topological indicesof certain networksrdquo Applied Mathematics and Computationvol 240 pp 213ndash228 2014

[8] S Hayat and M Imran ldquoComputation of certain topologicalindices of nanotubes covered byrdquo Journal of Computationaland eoretical Nanoscience vol 12 no 4 pp 533ndash541 2015

[9] A Q Baig M Imran and H Ali ldquoOn topological indices ofpoly oxide poly silicate DOX and DSL networksrdquo CanadianJournal of Chemistry vol 93 no 7 pp 730ndash739 2015

[10] Y Bashir A Aslam M Kamran et al ldquoOn forgotten topo-logical indices of some dendrimers structurerdquo Moleculesvol 22 no 6 p 867 2017

[11] A Aslam J L Guirao S Ahmad and W Gao ldquoTopologicalindices of the line graph of subdivision graph of completebipartite graphsrdquo Applied Mathematics amp Information Sci-ences vol 11 no 6 pp 1631ndash1636 2017

[12] G C Marsden P J Marchand P Harvey and S C EsenerldquoOptical transpose interconnection system architecturesrdquoOptics Letters vol 18 no 13 pp 1083ndash1085 1993

[13] A Al-Ayyoub A Awwad K Day and M Ould-KhaoualdquoGeneralized methods for algorithm development on opticalsystemsrdquo e Journal of Supercomputing vol 38 no 2pp 111ndash125 2006

[14] C F Wang and S Sahni ldquoMatrix multiplication on the OTIS-mesh optoelectronic computerrdquo IEEE Transactions onComputers vol 50 no 7 pp 635ndash646 2001

[15] C F Wang and S Sahni ldquoImage processing on the OTIS-mesh optoelectronic computerrdquo IEEE Transactions on Paralleland Distributed Systems vol 11 no 2 pp 97ndash109 2000

[16] C F Wang and S Sahni ldquoBasic operations on the OTIS-meshoptoelectronic computerrdquo IEEE Transactions on Parallel andDistributed Systems vol 9 no 12 pp 1226ndash1236 1998

10 Journal of Chemistry

[17] S Rajasekaran and S Sahni ldquoRandomized routing selectionand sorting on the OTIS-meshrdquo IEEE Transactions on Paralleland Distributed Systems vol 9 no 9 pp 833ndash840 1998

[18] A R Virk M N Jhangeer and M A Rehman ldquoReverseZagreb and reverse hyper-zagreb indices for silicon carbideSi2C3I[rs] and Si2C3II[rs]rdquo Engineering and Applied ScienceLetters vol 1 no 2 pp 37ndash50 2018

[19] N De ldquoComputing reformulated first Zagreb index of somechemical graphs as an application of generalized hierarchicalproduct of graphsrdquo Open Journal of Mathematical Sciencesvol 2 no 1 pp 338ndash350 2018

[20] N De ldquoHyper Zagreb index of bridge and chain graphsrdquoOpen Journal of Mathematical Sciences vol 2 no 1 pp 1ndash172018

[21] A T Balaban I Motoc D Bonchev and O MekenyanldquoTopological indices for structureactivity correlationsrdquo inSteric Effects in Drug Design pp 21ndash55 Springer BerlinGermany 1983

[22] I Gutman B Ruscic N Trinajstic and C F Wilcox JrldquoGraph theory and molecular orbitals XII Acyclic polyenesrdquoe Journal of Chemical Physics vol 62 no 9 pp 3399ndash34051975

[23] W Gao M Asif and W Nazeer ldquoe study of honey combderived network via topological indicesrdquo Open Journal ofMathematical Analysis vol 2 no 2 pp 10ndash26 2018

[24] H Siddiqui and M R Farahani ldquoForgotten polynomial andforgotten index of certain interconnection networksrdquo OpenJournal of Mathematical Analysis vol 1 no 1 pp 44ndash592017

[25] W Gao B Muzaffar and W Nazeer ldquoK-Banhatti and K-hyper Banhatti indices of dominating David derived net-workrdquo Open Journal of Mathematical Analysis vol 1 no 1pp 13ndash24 2017

[26] W Gao A Asghar andW Nazeer ldquoComputing degree-basedtopological indices of jahangir graphrdquo Engineering and Ap-plied Science Letters vol 1 no 1 pp 16ndash22 2018

[27] T Doslic B Furtula A Graovac I Gutman S Moradi andZ Yarahmadi ldquoOn vertex degree based molecular structuredescriptorsrdquo MATCH Communications in Mathematical andin Computer Chemistry vol 66 no 2 pp 613ndash626 2011

[28] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975

[29] M Randic ldquoOn history of the Randic index and emerginghostility toward chemical graph theoryrdquo MATCH Commu-nications in Mathematical and in Computer Chemistryvol 59 pp 5ndash124 2008

[30] M Randic ldquoe connectivity index 25 years afterrdquo Journal ofMolecular Graphics and Modelling vol 20 no 1 pp 19ndash352001

[31] K C Das and I Gutman ldquoSome properties of the secondZagreb indexrdquo MATCH Communications in Mathematicaland in Computer Chemistry vol 52 no 1 pp 3ndash1 2004

[32] L B Kier and L H HallMolecular Connectivity in Chemistryand Drug Research Academic Press New York NY USA1976

[33] L B Kier and L H HallMolecular Connectivity in Structure-Activity Analysis Research Studies Boston MA USA 1986

[34] X Li and I Gutman Mathematical Aspects of Randic-TypeMolecular Structure Descriptors University of KragujevacKragujevac Serbia 2006

[35] B Bollobas and P Erdos ldquoGraphs of extremal weightsrdquo ArsCombinatoria vol 50 pp 225ndash233 1998

[36] Y Hu X Li Y Shi T Xu and I Gutman ldquoOn moleculargraphs with smallest and greatest zeroth-order general Randicindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 54 no 2 pp 425ndash434 2005

[37] Y Ma S Cao Y Shi I Gutman M Dehmer and B FurtulaldquoFrom the connectivity index to various Randic-type de-scriptorsrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 80 no 1 pp 85ndash106 2018

[38] I Gutman B Furtula and S B Bozkurt ldquoOn Randic energyrdquoLinear Algebra and its Applications vol 442 pp 50ndash57 2014

[39] X Li and Y Shi ldquoA survey on the Randic indexrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 59 no 1 pp 127ndash156 2008

[40] M Cavers S Fallat and S Kirkland ldquoOn the normalizedLaplacian energy and general Randic index R-1 of graphsrdquoLinear Algebra and its Applications vol 433 no 1 pp 172ndash190 2010

[41] S B Bozkurt A D Gungor I Gutman and A S CevikldquoRandic matrix and Randic energyrdquo MATCH Communica-tions in Mathematical and in Computer Chemistry vol 64pp 239ndash250 2010

[42] L Pogliani ldquoFrom molecular connectivity indices to semi-empirical connectivity terms recent trends in graph theo-retical descriptorsrdquo Chemical Reviews vol 100 no 10pp 3827ndash3858 2000

[43] J B Liu M Younas M Habib M Yousaf and W NazeerldquoM-polynomials and degree-based topological indices ofVC5C7[p q] and HC5C7[p q] nanotubesrdquo IEEE Accessvol 7 pp 41125ndash41132 2019

[44] M Riaz W Gao and A Q Baig ldquoM-polynomials and degree-based topological indices of some families of convex poly-topesrdquo Open Journal of Mathematical Sciences vol 2 no 1pp 18ndash28 2018

[45] M S Anjum and M U Safdar ldquoK Banhatti and K hyper-Banhatti indices of nanotubesrdquo Engineering and AppliedScience Letters vol 2 no 1 pp 19ndash37 2019

[46] Z Shao A R Virk M S Javed M A Rehman andM R Farahani ldquoDegree based graph invariants for themolecular graph of Bismuth Tri-Iodiderdquo Engineering andApplied Science Letters vol 2 no 1 pp 01ndash11 2019

[47] L Yan M R Farahani and W Gao ldquoDistance-based indicescomputation of symmetry molecular structuresrdquo OpenJournal of Mathematical Sciences vol 2 no 1 pp 323ndash3372018

[48] R Kanabur and S K Hosamani ldquoSome numerical invariantsassociated with V-phenylenic nanotube and nanotorirdquo En-gineering and Applied Science Letters vol 1 no 1 pp 1ndash92018

[49] J B Liu C Wang and S Wang ldquoZagreb indices and mul-tiplicative Zagreb indices of eulerian graphsrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 42 pp 67ndash782019

[50] J B Liu and X F Pan ldquoAsymptotic Laplacian-energy-likeinvariant of latticesrdquo Applied Mathematics and Computationvol 253 pp 205ndash214 2015

[51] J B Liu and X-F Pan ldquoMinimizing Kirchhoffindex amonggraphs with a given vertex bipartitenessrdquo Applied Mathe-matics and Computation vol 291 pp 84ndash88 2016

[52] J B Liu J Zhao and Z Zhu ldquoOn the number of spanningtrees and normalized Laplacian of linear octagonal-quadri-lateral networksrdquo International Journal of Quantum Chem-istry vol 2019 Article ID e25971 2019

Journal of Chemistry 11

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Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom

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Volume 2018

Bioinorganic Chemistry and ApplicationsHindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

en

Dαy(f(x y)) 6αxy

3+ 6αx

2y2

+ 3α(6n minus 14)x2y3

+9α(n minus 2)(n minus 3)

2α1113888 1113889x

3y3

DαxD

αy1113888(f(x y)) 6αxy

3+ 6α2αx

2y2

+ 3α2α(6n minus 14)x2y3

+9α3α(n minus 2)(n minus 3)

2α1113888 1113889x

3y31113889

(26)

Now we have

Rα OPn( 1113857 DαxD

αyf|xy11113872

6α + 6α2α + 3α2α(6n minus 14) +9α3α(n minus 2)(n minus 3)

2α

(27)

Corollary 5 Let OPn be the swapped network then

RRα OPn( 1113857 2α

3α+3α

22α+6n minus 143α2α

+(n minus 2)(n minus 3)

2α3α1113888 1113889

(28)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(29)

en

Sαy(f(x y))

2α

3α1113888 1113889xy

3+

3α

2α1113888 1113889x

2y2

+(6n minus 14)

3α1113888 1113889x

2y3

+(n minus 2)(n minus 3)

2α1113888 1113889x

3y3

SαxS

αy1113888(f(x y))

2α

3α1113888 1113889xy

3+

3α

22α1113888 1113889x

2y2

+(6n minus 14)

3α2α1113888 1113889x

2y3

+(n minus 2)(n minus 3)

2α3α1113888 1113889x

3y31113889

(30)

Now we have

RRα OPn( 1113857 SαxS

αyf|xy11113872

2α

3α+3α

22α+6n minus 143α2α

+(n minus 2)(n minus 3)

2α3α

(31)

Corollary 6 Let OPn be the swapped network then

SSD OPn( 1113857 3n2

minus 2n +13 (32)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(33)

en

SxDy(f(x y)) 6xy3

+ 3x2y2

+3(6n minus 14)

21113888 1113889x

2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

DxSy(f(x y)) 23

1113874 1113875xy3

+ 3x2y2

+2(6n minus 14)

31113888 1113889x

2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(34)

Now we have

SSD OPn( 1113857 DxSyf + SxDyf1113872 111387311138681113868111386811138681113868xy1

3n2

minus 2n +13

(35)

Corollary 7 Let OPn be the swapped network then

H OPn( 1113857 2n2

4minus

n

20minus

120

1113888 1113889 (36)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(37)

en

J(f(x y)) 5x4

+(6n minus 14)x5

+3(n minus 2)(n minus 3)

21113888 1113889x

6

2SxJ(f(x y)) 25x4

4+

(6n minus 14)x5

5+

(n minus 2)(n minus 3)x6

41113888 1113889

(38)

Journal of Chemistry 5

Now we have

H OPn( 1113857 2SxJf1113868111386811138681113868x1

2n2

4minus

n

20minus

120

1113888 1113889

(39)

Corollary 8 Let OPn be the swapped network then

I OPn( 1113857 9n2

4minus81n

20minus1295

(40)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(41)

en

Dy(f(x y)) 6xy3

+ 6x2y2

+ 3(6n minus 14)x2y3

+9(n minus 2)(n minus 3)

21113888 1113889x

3y3

DxDy(f(x y)) 6xy3

+ 12x2y2

+ 6(6n minus 14)x2y3

+27(n minus 2)(n minus 3)

21113888 1113889x

3y3

JDxDy(f(x y)) 18x4

+ 6(6n minus 14)x5

+27(n minus 2)(n minus 3)

21113888 1113889x

6

SxJDxDy(f(x y)) 18x4

4+

6(6n minus 14)

51113888 1113889x

5

+27(n minus 2)(n minus 3)

121113888 1113889x

6

(42)

Now we have

I OPn( 1113857 SxJDxDyf11138681113868111386811138681113868x1

9n2

4minus81n

20minus1295

(43)

Corollary 9 Let OPn be the swapped network then

A OPn( 1113857 729n2

64minus573n

64minus41332

(44)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(45)

enD

3y(f(x y)) 54xy

3+ 24x

2y2

+ 27(6n minus 14)x2y3

+81(n minus 2)(n minus 3)

21113888 1113889x

3y3

D3xD

3y(f(x y)) 54xy

3+ 192x

2y2

+ 216(6n minus 14)x2y3

+2187(n minus 2)(n minus 3)

21113888 1113889x

3y3

JD3xD

3y(f(x y)) 246x

4+ 216(6n minus 14)x

5

+2187(n minus 2)(n minus 3)

21113888 1113889x

6

Qminus 2JD3xD

3y(f(x y)) 246x

2+ 216(6n minus 14)x

3

+2187(n minus 2)(n minus 3)

21113888 1113889x

4

S3xQminus 2JD

3xD

3y(f(x y))

123x2

4+ 8(6n minus 14)x

3

+729(n minus 2)(n minus 3)

641113888 1113889x

4

(46)Now we have

A OPn( 1113857 S3xQminus 2JD

3xD

3yf

11138681113868111386811138681113868x1

729n2

64minus573n

64minus41332

(47)

52 Results for (ORk)

Theorem 2 Let ORk be the swapped network then

M(G x y) nkxky

k+1+

n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(48)

Proof e ORk network has the following two types ofedges based on the degree of end vertices

E kk+1 ORk( 1113857 uv isin E ORk( 1113857 du k dv k + 11113864 1113865

E k+1k+1 ORk( 1113857 uv isin E ORk( 1113857 du k + 1 dv k + 11113864 1113865

(49)

6 Journal of Chemistry

such that

E kk+1 ORk( 11138571113868111386811138681113868

1113868111386811138681113868 nk

E k+1k+1 ORk( 11138571113868111386811138681113868

1113868111386811138681113868 n2(k + 1) minus n(1 + 2k)

2

(50)

Now from the definition of M-polynomial we have

M(G x y) 1113944δleilejleΔ

mijxiy

j

1113944klek+1

mk(k+1)xky

k+1+ 1113944

k+1lek+1m(k+1)(k+1)x

k+1y

k+1

E kk+1 ORk( 11138571113868111386811138681113868

1113868111386811138681113868xky

k+1+ E k+1k+1 ORk( 1113857

11138681113868111386811138681113868111386811138681113868x

k+1y

k+1

nkxky

k+1+

n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(51)

Corollary 10 Let (ORk) be the swapped network then

M1 ORk( 1113857 2nk2

+ nk + n2(k + 1)

2minus n(k + 1)(1 + 2k)

(52)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(53)

en

Dx(f(x y)) nk2x

ky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

Dy(f(x y)) nk(k + 1)xky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(54)

Now we have

M1 ORk( 1113857 Dxf + Dyf11138681113868111386811138681113868xy1

2nk2

+ nk + n2(k + 1)

2minus n(k + 1)(1 + 2k)

(55)

Corollary 11 Let (ORk) be the swapped network then

M2 ORk( 1113857 nk3

+ nk2

+n2(k + 1)3

2minus

n(k + 1)2(1 + 2k)

2

(56)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(57)

en

Dy(f(x y)) nk2x

ky

k+1+ nkx

ky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxDy(f(x y)) nk3x

ky

k+1+ nk

2x

ky

k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(58)

Now we have

M2 ORk( 1113857 DxDyf11138681113868111386811138681113868xy1

nk3

+ nk2

+n2(k + 1)3

2minus

n(k + 1)2(1 + 2k)

2

(59)

Corollary 12 Let (ORk) be the swapped network then

mM2 ORk( 1113857

nk

k2 + k( )+

n2

(2k + 2)minus

n(1 + 2k)

2(k + 1)2 (60)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(61)

en

Sy(f(x y)) nk

(k + 1)1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2(k + 1)1113888 1113889x

k+1y

k+1

SxSy(f(x y)) nk

k2 + k( )1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2(k + 1)21113888 1113889x

k+1y

k+1

(62)

Journal of Chemistry 7

Now we havem

M2 ORk( 1113857 SxSyf11138681113868111386811138681113868xy1

nk

k2 + k( )+

n2

(2k + 2)minus

n(1 + 2k)

2(k + 1)2

(63)

Corollary 13 Let (ORk) be the swapped network then

Rα1113888 ORk( 1113857 nk3α

+ nk2α

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

2α1113888 1113889x

k+1y

k+11113889

(64)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(65)

en

Dαy(f(x y)) nk

2αx

ky

k+1+ nk

αx

ky

k+1

+(k + 1)α n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DαxD

αy1113888(f(x y)) nk

3αx

ky

k+1+ nk

2αx

ky

k+1

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+11113889

(66)

Now we have

Rα ORk( 1113857 DαxD

αyf

11138681113868111386811138681113868xy11113874

nk3α

+ nk2α

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

2α1113888 1113889x

k+1y

k+1

(67)

Corollary 14 Let (ORk) be the swapped network then

RRα ORk( 1113857 n

(k + 1)α+

n2(k + 1) minus n(1 + 2k)

(k + 1)2α2α1113888 11138891113888 1113889

(68)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(69)

en

Sαy(f(x y))

nkα

(k + 1)α1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2α(k + 1)α1113888 1113889x

k+1y

k+1

SαxS

αy1113888(f(x y))

n

(k + 1)α1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2α(k + 1)2α1113888 1113889x

k+1y

k+11113889

(70)

Now we have

RRα ORk( 1113857 SαxS

αyf

11138681113868111386811138681113868xy11113874

n

(k + 1)α+

n2(k + 1) minus n(1 + 2k)

(k + 1)2α2α1113888 1113889

(71)

Corollary 15 Let (ORk) be the swapped network then

SSD ORk( 1113857 nk2

(k + 1)+ nk + n + n

2(k + 1) minus n(1 + 2k)

(72)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(73)

en

SxDy(f(x y)) nkxky

k+1+ nx

ky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxSy(f(x y)) nk2

(k + 1)1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(74)

Now we have

SSD ORk( 1113857 DxSyf + SxDyf1113872 111387311138681113868111386811138681113868xy1

nk2

(k + 1)+ nk + n + n

2(k + 1) minus n(1 + 2k)

(75)

8 Journal of Chemistry

Corollary 16 Let (ORk) be the swapped network then

H ORk( 1113857 2nk

2k + 1+

n2(k + 1)

4k + 4minus

n(1 + 2k)

4k + 41113888 1113889 (76)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(77)

en

J(f(x y)) nkx2k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

2SxJ(f(x y)) 21113888nk

2k + 11113888 1113889x

2k+1

+n2(k + 1)

4k + 41113888 1113889x

2k+2

minusn(1 + 2k)

4k + 41113888 1113889x

2k+21113889

(78)

Now we have

H ORk( 1113857 2SxJf1113868111386811138681113868x1

2nk

2k + 1+

n2(k + 1)

4k + 4minus

n(1 + 2k)

4k + 41113888 1113889

(79)

Corollary 17 Let (ORk) be the swapped network then

I ORk( 1113857 nk3

2k + 1+

nk2

2k + 1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889

(80)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(81)

en

Dy(f(x y)) nk(k + 1)xky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxDy(f(x y)) nk3x

ky

k+1+ nk

2x

ky

k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

JDxDy(f(x y)) nk3x2k+1

+ nk2x2k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

SxJDxDy(f(x y)) nk3

2k + 11113888 1113889x

2k+1+

nk2

2k + 11113888 1113889x

2k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889x

2k+2

(82)

Now we have

I ORk( 1113857 SxJDxDyf11138681113868111386811138681113868x1

nk3

2k + 1+

nk2

2k + 1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889

(83)

Corollary 18 Let (ORk) be the swapped network then

A ORk( 1113857 nk5(k + 1)2

(2k minus 1)3

+nk4(k + 1)2

(2k minus 1)3

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889

(84)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(85)

Journal of Chemistry 9

en

D3y(f(x y)) nk

2(k + 1)

2x

ky

k+1+ nk(k + 1)

2x

ky

k+1

+(k + 1)3 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

D3yD

3y(f(x y)) nk

5(k + 1)

2x

ky

k+1+ nk

4(k + 1)

2x

ky

k+1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

JD3yD

3y(f(x y)) nk

5(k + 1)

2x2k+1

+ nk4(k + 1)

2x2k+1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

Qminus 2JD3yD

3y(f(x y)) nk

5(k + 1)

2x2kminus 1

+ nk4(k + 1)

2x2kminus 1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k

S3xQminus 2JD

3yD

3y(f(x y))

nk5(k + 1)2

(2k minus 1)31113888 1113889x

2kminus 1+

nk4(k + 1)2

(2k minus 1)31113888 1113889x

2kminus 1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889x2k

(86)

Now we have

A ORk( 1113857 S3xQminus 2JD

3yD

3yf

11138681113868111386811138681113868x1

nk5(k + 1)2

(2k minus 1)3+

nk4(k + 1)2

(2k minus 1)3

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889

(87)

6 Conclusion

In this paper our focus is on swapped interconnectionnetworks that allow systematic construction of large scal-able modular and robust parallel architectures whilemaintaining many desirable attributes of the underlyingbasis network comprising its clusters We have computedseveral TIs of underlined networks Firstly we computed M-polynomials of understudy networks and then we recoveredZagreb indices Randic indices and some other indices[49ndash52]

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to this study

Acknowledgments

is study was partially supported by the Natural ScienceFoundation of Anhui Province (no KJ2018A059)

References

[1] W Gao M R Farahani and L Shi ldquoForgotten topologicalindex of some drug structuresrdquo Acta Medica Mediterraneavol 32 no 1 pp 579ndash585 2016

[2] W Gao M K Siddiqui M Imran M K Jamil andM R Farahani ldquoForgotten topological index of chemicalstructure in drugsrdquo Saudi Pharmaceutical Journal vol 24no 3 pp 258ndash264 2016

[3] W Gao W Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 Article ID 3216327 8 pages 2016

[4] M Imran A Asghar and A Q Baig ldquoOn graph invariants ofoxide networkrdquo Engineering and Applied Science Lettersvol 1(2018) no 1 pp 23ndash28 2018

[5] G Liu Z Jia and W Gao ldquoOntology similarity computingbased on stochastic primal dual coordinate techniquerdquo OpenJournal of Mathematical Sciences vol 2(2018) no 1pp 221ndash227 2018

[6] S Noreen and A Mahmood ldquoZagreb polynomials andredefined Zagreb indices for the line graph of carbonnanoconesrdquo Open Journal of Mathematical Analysisvol 2(2018) no 1 pp 66ndash73 2018

[7] S Hayat and M Imran ldquoComputation of topological indicesof certain networksrdquo Applied Mathematics and Computationvol 240 pp 213ndash228 2014

[8] S Hayat and M Imran ldquoComputation of certain topologicalindices of nanotubes covered byrdquo Journal of Computationaland eoretical Nanoscience vol 12 no 4 pp 533ndash541 2015

[9] A Q Baig M Imran and H Ali ldquoOn topological indices ofpoly oxide poly silicate DOX and DSL networksrdquo CanadianJournal of Chemistry vol 93 no 7 pp 730ndash739 2015

[10] Y Bashir A Aslam M Kamran et al ldquoOn forgotten topo-logical indices of some dendrimers structurerdquo Moleculesvol 22 no 6 p 867 2017

[11] A Aslam J L Guirao S Ahmad and W Gao ldquoTopologicalindices of the line graph of subdivision graph of completebipartite graphsrdquo Applied Mathematics amp Information Sci-ences vol 11 no 6 pp 1631ndash1636 2017

[12] G C Marsden P J Marchand P Harvey and S C EsenerldquoOptical transpose interconnection system architecturesrdquoOptics Letters vol 18 no 13 pp 1083ndash1085 1993

[13] A Al-Ayyoub A Awwad K Day and M Ould-KhaoualdquoGeneralized methods for algorithm development on opticalsystemsrdquo e Journal of Supercomputing vol 38 no 2pp 111ndash125 2006

[14] C F Wang and S Sahni ldquoMatrix multiplication on the OTIS-mesh optoelectronic computerrdquo IEEE Transactions onComputers vol 50 no 7 pp 635ndash646 2001

[15] C F Wang and S Sahni ldquoImage processing on the OTIS-mesh optoelectronic computerrdquo IEEE Transactions on Paralleland Distributed Systems vol 11 no 2 pp 97ndash109 2000

[16] C F Wang and S Sahni ldquoBasic operations on the OTIS-meshoptoelectronic computerrdquo IEEE Transactions on Parallel andDistributed Systems vol 9 no 12 pp 1226ndash1236 1998

10 Journal of Chemistry

[17] S Rajasekaran and S Sahni ldquoRandomized routing selectionand sorting on the OTIS-meshrdquo IEEE Transactions on Paralleland Distributed Systems vol 9 no 9 pp 833ndash840 1998

[18] A R Virk M N Jhangeer and M A Rehman ldquoReverseZagreb and reverse hyper-zagreb indices for silicon carbideSi2C3I[rs] and Si2C3II[rs]rdquo Engineering and Applied ScienceLetters vol 1 no 2 pp 37ndash50 2018

[19] N De ldquoComputing reformulated first Zagreb index of somechemical graphs as an application of generalized hierarchicalproduct of graphsrdquo Open Journal of Mathematical Sciencesvol 2 no 1 pp 338ndash350 2018

[20] N De ldquoHyper Zagreb index of bridge and chain graphsrdquoOpen Journal of Mathematical Sciences vol 2 no 1 pp 1ndash172018

[21] A T Balaban I Motoc D Bonchev and O MekenyanldquoTopological indices for structureactivity correlationsrdquo inSteric Effects in Drug Design pp 21ndash55 Springer BerlinGermany 1983

[22] I Gutman B Ruscic N Trinajstic and C F Wilcox JrldquoGraph theory and molecular orbitals XII Acyclic polyenesrdquoe Journal of Chemical Physics vol 62 no 9 pp 3399ndash34051975

[23] W Gao M Asif and W Nazeer ldquoe study of honey combderived network via topological indicesrdquo Open Journal ofMathematical Analysis vol 2 no 2 pp 10ndash26 2018

[24] H Siddiqui and M R Farahani ldquoForgotten polynomial andforgotten index of certain interconnection networksrdquo OpenJournal of Mathematical Analysis vol 1 no 1 pp 44ndash592017

[25] W Gao B Muzaffar and W Nazeer ldquoK-Banhatti and K-hyper Banhatti indices of dominating David derived net-workrdquo Open Journal of Mathematical Analysis vol 1 no 1pp 13ndash24 2017

[26] W Gao A Asghar andW Nazeer ldquoComputing degree-basedtopological indices of jahangir graphrdquo Engineering and Ap-plied Science Letters vol 1 no 1 pp 16ndash22 2018

[27] T Doslic B Furtula A Graovac I Gutman S Moradi andZ Yarahmadi ldquoOn vertex degree based molecular structuredescriptorsrdquo MATCH Communications in Mathematical andin Computer Chemistry vol 66 no 2 pp 613ndash626 2011

[28] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975

[29] M Randic ldquoOn history of the Randic index and emerginghostility toward chemical graph theoryrdquo MATCH Commu-nications in Mathematical and in Computer Chemistryvol 59 pp 5ndash124 2008

[30] M Randic ldquoe connectivity index 25 years afterrdquo Journal ofMolecular Graphics and Modelling vol 20 no 1 pp 19ndash352001

[31] K C Das and I Gutman ldquoSome properties of the secondZagreb indexrdquo MATCH Communications in Mathematicaland in Computer Chemistry vol 52 no 1 pp 3ndash1 2004

[32] L B Kier and L H HallMolecular Connectivity in Chemistryand Drug Research Academic Press New York NY USA1976

[33] L B Kier and L H HallMolecular Connectivity in Structure-Activity Analysis Research Studies Boston MA USA 1986

[34] X Li and I Gutman Mathematical Aspects of Randic-TypeMolecular Structure Descriptors University of KragujevacKragujevac Serbia 2006

[35] B Bollobas and P Erdos ldquoGraphs of extremal weightsrdquo ArsCombinatoria vol 50 pp 225ndash233 1998

[36] Y Hu X Li Y Shi T Xu and I Gutman ldquoOn moleculargraphs with smallest and greatest zeroth-order general Randicindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 54 no 2 pp 425ndash434 2005

[37] Y Ma S Cao Y Shi I Gutman M Dehmer and B FurtulaldquoFrom the connectivity index to various Randic-type de-scriptorsrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 80 no 1 pp 85ndash106 2018

[38] I Gutman B Furtula and S B Bozkurt ldquoOn Randic energyrdquoLinear Algebra and its Applications vol 442 pp 50ndash57 2014

[39] X Li and Y Shi ldquoA survey on the Randic indexrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 59 no 1 pp 127ndash156 2008

[40] M Cavers S Fallat and S Kirkland ldquoOn the normalizedLaplacian energy and general Randic index R-1 of graphsrdquoLinear Algebra and its Applications vol 433 no 1 pp 172ndash190 2010

[41] S B Bozkurt A D Gungor I Gutman and A S CevikldquoRandic matrix and Randic energyrdquo MATCH Communica-tions in Mathematical and in Computer Chemistry vol 64pp 239ndash250 2010

[42] L Pogliani ldquoFrom molecular connectivity indices to semi-empirical connectivity terms recent trends in graph theo-retical descriptorsrdquo Chemical Reviews vol 100 no 10pp 3827ndash3858 2000

[43] J B Liu M Younas M Habib M Yousaf and W NazeerldquoM-polynomials and degree-based topological indices ofVC5C7[p q] and HC5C7[p q] nanotubesrdquo IEEE Accessvol 7 pp 41125ndash41132 2019

[44] M Riaz W Gao and A Q Baig ldquoM-polynomials and degree-based topological indices of some families of convex poly-topesrdquo Open Journal of Mathematical Sciences vol 2 no 1pp 18ndash28 2018

[45] M S Anjum and M U Safdar ldquoK Banhatti and K hyper-Banhatti indices of nanotubesrdquo Engineering and AppliedScience Letters vol 2 no 1 pp 19ndash37 2019

[46] Z Shao A R Virk M S Javed M A Rehman andM R Farahani ldquoDegree based graph invariants for themolecular graph of Bismuth Tri-Iodiderdquo Engineering andApplied Science Letters vol 2 no 1 pp 01ndash11 2019

[47] L Yan M R Farahani and W Gao ldquoDistance-based indicescomputation of symmetry molecular structuresrdquo OpenJournal of Mathematical Sciences vol 2 no 1 pp 323ndash3372018

[48] R Kanabur and S K Hosamani ldquoSome numerical invariantsassociated with V-phenylenic nanotube and nanotorirdquo En-gineering and Applied Science Letters vol 1 no 1 pp 1ndash92018

[49] J B Liu C Wang and S Wang ldquoZagreb indices and mul-tiplicative Zagreb indices of eulerian graphsrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 42 pp 67ndash782019

[50] J B Liu and X F Pan ldquoAsymptotic Laplacian-energy-likeinvariant of latticesrdquo Applied Mathematics and Computationvol 253 pp 205ndash214 2015

[51] J B Liu and X-F Pan ldquoMinimizing Kirchhoffindex amonggraphs with a given vertex bipartitenessrdquo Applied Mathe-matics and Computation vol 291 pp 84ndash88 2016

[52] J B Liu J Zhao and Z Zhu ldquoOn the number of spanningtrees and normalized Laplacian of linear octagonal-quadri-lateral networksrdquo International Journal of Quantum Chem-istry vol 2019 Article ID e25971 2019

Journal of Chemistry 11

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nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

Now we have

H OPn( 1113857 2SxJf1113868111386811138681113868x1

2n2

4minus

n

20minus

120

1113888 1113889

(39)

Corollary 8 Let OPn be the swapped network then

I OPn( 1113857 9n2

4minus81n

20minus1295

(40)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(41)

en

Dy(f(x y)) 6xy3

+ 6x2y2

+ 3(6n minus 14)x2y3

+9(n minus 2)(n minus 3)

21113888 1113889x

3y3

DxDy(f(x y)) 6xy3

+ 12x2y2

+ 6(6n minus 14)x2y3

+27(n minus 2)(n minus 3)

21113888 1113889x

3y3

JDxDy(f(x y)) 18x4

+ 6(6n minus 14)x5

+27(n minus 2)(n minus 3)

21113888 1113889x

6

SxJDxDy(f(x y)) 18x4

4+

6(6n minus 14)

51113888 1113889x

5

+27(n minus 2)(n minus 3)

121113888 1113889x

6

(42)

Now we have

I OPn( 1113857 SxJDxDyf11138681113868111386811138681113868x1

9n2

4minus81n

20minus1295

(43)

Corollary 9 Let OPn be the swapped network then

A OPn( 1113857 729n2

64minus573n

64minus41332

(44)

Proof Let

f(x y) M(G x y) 2xy3

+ 3x2y2

+(6n minus 14)x2y3

+3(n minus 2)(n minus 3)

21113888 1113889x

3y3

(45)

enD

3y(f(x y)) 54xy

3+ 24x

2y2

+ 27(6n minus 14)x2y3

+81(n minus 2)(n minus 3)

21113888 1113889x

3y3

D3xD

3y(f(x y)) 54xy

3+ 192x

2y2

+ 216(6n minus 14)x2y3

+2187(n minus 2)(n minus 3)

21113888 1113889x

3y3

JD3xD

3y(f(x y)) 246x

4+ 216(6n minus 14)x

5

+2187(n minus 2)(n minus 3)

21113888 1113889x

6

Qminus 2JD3xD

3y(f(x y)) 246x

2+ 216(6n minus 14)x

3

+2187(n minus 2)(n minus 3)

21113888 1113889x

4

S3xQminus 2JD

3xD

3y(f(x y))

123x2

4+ 8(6n minus 14)x

3

+729(n minus 2)(n minus 3)

641113888 1113889x

4

(46)Now we have

A OPn( 1113857 S3xQminus 2JD

3xD

3yf

11138681113868111386811138681113868x1

729n2

64minus573n

64minus41332

(47)

52 Results for (ORk)

Theorem 2 Let ORk be the swapped network then

M(G x y) nkxky

k+1+

n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(48)

Proof e ORk network has the following two types ofedges based on the degree of end vertices

E kk+1 ORk( 1113857 uv isin E ORk( 1113857 du k dv k + 11113864 1113865

E k+1k+1 ORk( 1113857 uv isin E ORk( 1113857 du k + 1 dv k + 11113864 1113865

(49)

6 Journal of Chemistry

such that

E kk+1 ORk( 11138571113868111386811138681113868

1113868111386811138681113868 nk

E k+1k+1 ORk( 11138571113868111386811138681113868

1113868111386811138681113868 n2(k + 1) minus n(1 + 2k)

2

(50)

Now from the definition of M-polynomial we have

M(G x y) 1113944δleilejleΔ

mijxiy

j

1113944klek+1

mk(k+1)xky

k+1+ 1113944

k+1lek+1m(k+1)(k+1)x

k+1y

k+1

E kk+1 ORk( 11138571113868111386811138681113868

1113868111386811138681113868xky

k+1+ E k+1k+1 ORk( 1113857

11138681113868111386811138681113868111386811138681113868x

k+1y

k+1

nkxky

k+1+

n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(51)

Corollary 10 Let (ORk) be the swapped network then

M1 ORk( 1113857 2nk2

+ nk + n2(k + 1)

2minus n(k + 1)(1 + 2k)

(52)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(53)

en

Dx(f(x y)) nk2x

ky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

Dy(f(x y)) nk(k + 1)xky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(54)

Now we have

M1 ORk( 1113857 Dxf + Dyf11138681113868111386811138681113868xy1

2nk2

+ nk + n2(k + 1)

2minus n(k + 1)(1 + 2k)

(55)

Corollary 11 Let (ORk) be the swapped network then

M2 ORk( 1113857 nk3

+ nk2

+n2(k + 1)3

2minus

n(k + 1)2(1 + 2k)

2

(56)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(57)

en

Dy(f(x y)) nk2x

ky

k+1+ nkx

ky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxDy(f(x y)) nk3x

ky

k+1+ nk

2x

ky

k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(58)

Now we have

M2 ORk( 1113857 DxDyf11138681113868111386811138681113868xy1

nk3

+ nk2

+n2(k + 1)3

2minus

n(k + 1)2(1 + 2k)

2

(59)

Corollary 12 Let (ORk) be the swapped network then

mM2 ORk( 1113857

nk

k2 + k( )+

n2

(2k + 2)minus

n(1 + 2k)

2(k + 1)2 (60)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(61)

en

Sy(f(x y)) nk

(k + 1)1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2(k + 1)1113888 1113889x

k+1y

k+1

SxSy(f(x y)) nk

k2 + k( )1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2(k + 1)21113888 1113889x

k+1y

k+1

(62)

Journal of Chemistry 7

Now we havem

M2 ORk( 1113857 SxSyf11138681113868111386811138681113868xy1

nk

k2 + k( )+

n2

(2k + 2)minus

n(1 + 2k)

2(k + 1)2

(63)

Corollary 13 Let (ORk) be the swapped network then

Rα1113888 ORk( 1113857 nk3α

+ nk2α

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

2α1113888 1113889x

k+1y

k+11113889

(64)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(65)

en

Dαy(f(x y)) nk

2αx

ky

k+1+ nk

αx

ky

k+1

+(k + 1)α n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DαxD

αy1113888(f(x y)) nk

3αx

ky

k+1+ nk

2αx

ky

k+1

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+11113889

(66)

Now we have

Rα ORk( 1113857 DαxD

αyf

11138681113868111386811138681113868xy11113874

nk3α

+ nk2α

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

2α1113888 1113889x

k+1y

k+1

(67)

Corollary 14 Let (ORk) be the swapped network then

RRα ORk( 1113857 n

(k + 1)α+

n2(k + 1) minus n(1 + 2k)

(k + 1)2α2α1113888 11138891113888 1113889

(68)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(69)

en

Sαy(f(x y))

nkα

(k + 1)α1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2α(k + 1)α1113888 1113889x

k+1y

k+1

SαxS

αy1113888(f(x y))

n

(k + 1)α1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2α(k + 1)2α1113888 1113889x

k+1y

k+11113889

(70)

Now we have

RRα ORk( 1113857 SαxS

αyf

11138681113868111386811138681113868xy11113874

n

(k + 1)α+

n2(k + 1) minus n(1 + 2k)

(k + 1)2α2α1113888 1113889

(71)

Corollary 15 Let (ORk) be the swapped network then

SSD ORk( 1113857 nk2

(k + 1)+ nk + n + n

2(k + 1) minus n(1 + 2k)

(72)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(73)

en

SxDy(f(x y)) nkxky

k+1+ nx

ky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxSy(f(x y)) nk2

(k + 1)1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(74)

Now we have

SSD ORk( 1113857 DxSyf + SxDyf1113872 111387311138681113868111386811138681113868xy1

nk2

(k + 1)+ nk + n + n

2(k + 1) minus n(1 + 2k)

(75)

8 Journal of Chemistry

Corollary 16 Let (ORk) be the swapped network then

H ORk( 1113857 2nk

2k + 1+

n2(k + 1)

4k + 4minus

n(1 + 2k)

4k + 41113888 1113889 (76)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(77)

en

J(f(x y)) nkx2k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

2SxJ(f(x y)) 21113888nk

2k + 11113888 1113889x

2k+1

+n2(k + 1)

4k + 41113888 1113889x

2k+2

minusn(1 + 2k)

4k + 41113888 1113889x

2k+21113889

(78)

Now we have

H ORk( 1113857 2SxJf1113868111386811138681113868x1

2nk

2k + 1+

n2(k + 1)

4k + 4minus

n(1 + 2k)

4k + 41113888 1113889

(79)

Corollary 17 Let (ORk) be the swapped network then

I ORk( 1113857 nk3

2k + 1+

nk2

2k + 1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889

(80)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(81)

en

Dy(f(x y)) nk(k + 1)xky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxDy(f(x y)) nk3x

ky

k+1+ nk

2x

ky

k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

JDxDy(f(x y)) nk3x2k+1

+ nk2x2k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

SxJDxDy(f(x y)) nk3

2k + 11113888 1113889x

2k+1+

nk2

2k + 11113888 1113889x

2k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889x

2k+2

(82)

Now we have

I ORk( 1113857 SxJDxDyf11138681113868111386811138681113868x1

nk3

2k + 1+

nk2

2k + 1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889

(83)

Corollary 18 Let (ORk) be the swapped network then

A ORk( 1113857 nk5(k + 1)2

(2k minus 1)3

+nk4(k + 1)2

(2k minus 1)3

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889

(84)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(85)

Journal of Chemistry 9

en

D3y(f(x y)) nk

2(k + 1)

2x

ky

k+1+ nk(k + 1)

2x

ky

k+1

+(k + 1)3 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

D3yD

3y(f(x y)) nk

5(k + 1)

2x

ky

k+1+ nk

4(k + 1)

2x

ky

k+1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

JD3yD

3y(f(x y)) nk

5(k + 1)

2x2k+1

+ nk4(k + 1)

2x2k+1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

Qminus 2JD3yD

3y(f(x y)) nk

5(k + 1)

2x2kminus 1

+ nk4(k + 1)

2x2kminus 1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k

S3xQminus 2JD

3yD

3y(f(x y))

nk5(k + 1)2

(2k minus 1)31113888 1113889x

2kminus 1+

nk4(k + 1)2

(2k minus 1)31113888 1113889x

2kminus 1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889x2k

(86)

Now we have

A ORk( 1113857 S3xQminus 2JD

3yD

3yf

11138681113868111386811138681113868x1

nk5(k + 1)2

(2k minus 1)3+

nk4(k + 1)2

(2k minus 1)3

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889

(87)

6 Conclusion

In this paper our focus is on swapped interconnectionnetworks that allow systematic construction of large scal-able modular and robust parallel architectures whilemaintaining many desirable attributes of the underlyingbasis network comprising its clusters We have computedseveral TIs of underlined networks Firstly we computed M-polynomials of understudy networks and then we recoveredZagreb indices Randic indices and some other indices[49ndash52]

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to this study

Acknowledgments

is study was partially supported by the Natural ScienceFoundation of Anhui Province (no KJ2018A059)

References

[1] W Gao M R Farahani and L Shi ldquoForgotten topologicalindex of some drug structuresrdquo Acta Medica Mediterraneavol 32 no 1 pp 579ndash585 2016

[2] W Gao M K Siddiqui M Imran M K Jamil andM R Farahani ldquoForgotten topological index of chemicalstructure in drugsrdquo Saudi Pharmaceutical Journal vol 24no 3 pp 258ndash264 2016

[3] W Gao W Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 Article ID 3216327 8 pages 2016

[4] M Imran A Asghar and A Q Baig ldquoOn graph invariants ofoxide networkrdquo Engineering and Applied Science Lettersvol 1(2018) no 1 pp 23ndash28 2018

[5] G Liu Z Jia and W Gao ldquoOntology similarity computingbased on stochastic primal dual coordinate techniquerdquo OpenJournal of Mathematical Sciences vol 2(2018) no 1pp 221ndash227 2018

[6] S Noreen and A Mahmood ldquoZagreb polynomials andredefined Zagreb indices for the line graph of carbonnanoconesrdquo Open Journal of Mathematical Analysisvol 2(2018) no 1 pp 66ndash73 2018

[7] S Hayat and M Imran ldquoComputation of topological indicesof certain networksrdquo Applied Mathematics and Computationvol 240 pp 213ndash228 2014

[8] S Hayat and M Imran ldquoComputation of certain topologicalindices of nanotubes covered byrdquo Journal of Computationaland eoretical Nanoscience vol 12 no 4 pp 533ndash541 2015

[9] A Q Baig M Imran and H Ali ldquoOn topological indices ofpoly oxide poly silicate DOX and DSL networksrdquo CanadianJournal of Chemistry vol 93 no 7 pp 730ndash739 2015

[10] Y Bashir A Aslam M Kamran et al ldquoOn forgotten topo-logical indices of some dendrimers structurerdquo Moleculesvol 22 no 6 p 867 2017

[11] A Aslam J L Guirao S Ahmad and W Gao ldquoTopologicalindices of the line graph of subdivision graph of completebipartite graphsrdquo Applied Mathematics amp Information Sci-ences vol 11 no 6 pp 1631ndash1636 2017

[12] G C Marsden P J Marchand P Harvey and S C EsenerldquoOptical transpose interconnection system architecturesrdquoOptics Letters vol 18 no 13 pp 1083ndash1085 1993

[13] A Al-Ayyoub A Awwad K Day and M Ould-KhaoualdquoGeneralized methods for algorithm development on opticalsystemsrdquo e Journal of Supercomputing vol 38 no 2pp 111ndash125 2006

[14] C F Wang and S Sahni ldquoMatrix multiplication on the OTIS-mesh optoelectronic computerrdquo IEEE Transactions onComputers vol 50 no 7 pp 635ndash646 2001

[15] C F Wang and S Sahni ldquoImage processing on the OTIS-mesh optoelectronic computerrdquo IEEE Transactions on Paralleland Distributed Systems vol 11 no 2 pp 97ndash109 2000

[16] C F Wang and S Sahni ldquoBasic operations on the OTIS-meshoptoelectronic computerrdquo IEEE Transactions on Parallel andDistributed Systems vol 9 no 12 pp 1226ndash1236 1998

10 Journal of Chemistry

[17] S Rajasekaran and S Sahni ldquoRandomized routing selectionand sorting on the OTIS-meshrdquo IEEE Transactions on Paralleland Distributed Systems vol 9 no 9 pp 833ndash840 1998

[18] A R Virk M N Jhangeer and M A Rehman ldquoReverseZagreb and reverse hyper-zagreb indices for silicon carbideSi2C3I[rs] and Si2C3II[rs]rdquo Engineering and Applied ScienceLetters vol 1 no 2 pp 37ndash50 2018

[19] N De ldquoComputing reformulated first Zagreb index of somechemical graphs as an application of generalized hierarchicalproduct of graphsrdquo Open Journal of Mathematical Sciencesvol 2 no 1 pp 338ndash350 2018

[20] N De ldquoHyper Zagreb index of bridge and chain graphsrdquoOpen Journal of Mathematical Sciences vol 2 no 1 pp 1ndash172018

[21] A T Balaban I Motoc D Bonchev and O MekenyanldquoTopological indices for structureactivity correlationsrdquo inSteric Effects in Drug Design pp 21ndash55 Springer BerlinGermany 1983

[22] I Gutman B Ruscic N Trinajstic and C F Wilcox JrldquoGraph theory and molecular orbitals XII Acyclic polyenesrdquoe Journal of Chemical Physics vol 62 no 9 pp 3399ndash34051975

[23] W Gao M Asif and W Nazeer ldquoe study of honey combderived network via topological indicesrdquo Open Journal ofMathematical Analysis vol 2 no 2 pp 10ndash26 2018

[24] H Siddiqui and M R Farahani ldquoForgotten polynomial andforgotten index of certain interconnection networksrdquo OpenJournal of Mathematical Analysis vol 1 no 1 pp 44ndash592017

[25] W Gao B Muzaffar and W Nazeer ldquoK-Banhatti and K-hyper Banhatti indices of dominating David derived net-workrdquo Open Journal of Mathematical Analysis vol 1 no 1pp 13ndash24 2017

[26] W Gao A Asghar andW Nazeer ldquoComputing degree-basedtopological indices of jahangir graphrdquo Engineering and Ap-plied Science Letters vol 1 no 1 pp 16ndash22 2018

[27] T Doslic B Furtula A Graovac I Gutman S Moradi andZ Yarahmadi ldquoOn vertex degree based molecular structuredescriptorsrdquo MATCH Communications in Mathematical andin Computer Chemistry vol 66 no 2 pp 613ndash626 2011

[28] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975

[29] M Randic ldquoOn history of the Randic index and emerginghostility toward chemical graph theoryrdquo MATCH Commu-nications in Mathematical and in Computer Chemistryvol 59 pp 5ndash124 2008

[30] M Randic ldquoe connectivity index 25 years afterrdquo Journal ofMolecular Graphics and Modelling vol 20 no 1 pp 19ndash352001

[31] K C Das and I Gutman ldquoSome properties of the secondZagreb indexrdquo MATCH Communications in Mathematicaland in Computer Chemistry vol 52 no 1 pp 3ndash1 2004

[32] L B Kier and L H HallMolecular Connectivity in Chemistryand Drug Research Academic Press New York NY USA1976

[33] L B Kier and L H HallMolecular Connectivity in Structure-Activity Analysis Research Studies Boston MA USA 1986

[34] X Li and I Gutman Mathematical Aspects of Randic-TypeMolecular Structure Descriptors University of KragujevacKragujevac Serbia 2006

[35] B Bollobas and P Erdos ldquoGraphs of extremal weightsrdquo ArsCombinatoria vol 50 pp 225ndash233 1998

[36] Y Hu X Li Y Shi T Xu and I Gutman ldquoOn moleculargraphs with smallest and greatest zeroth-order general Randicindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 54 no 2 pp 425ndash434 2005

[37] Y Ma S Cao Y Shi I Gutman M Dehmer and B FurtulaldquoFrom the connectivity index to various Randic-type de-scriptorsrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 80 no 1 pp 85ndash106 2018

[38] I Gutman B Furtula and S B Bozkurt ldquoOn Randic energyrdquoLinear Algebra and its Applications vol 442 pp 50ndash57 2014

[39] X Li and Y Shi ldquoA survey on the Randic indexrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 59 no 1 pp 127ndash156 2008

[40] M Cavers S Fallat and S Kirkland ldquoOn the normalizedLaplacian energy and general Randic index R-1 of graphsrdquoLinear Algebra and its Applications vol 433 no 1 pp 172ndash190 2010

[41] S B Bozkurt A D Gungor I Gutman and A S CevikldquoRandic matrix and Randic energyrdquo MATCH Communica-tions in Mathematical and in Computer Chemistry vol 64pp 239ndash250 2010

[42] L Pogliani ldquoFrom molecular connectivity indices to semi-empirical connectivity terms recent trends in graph theo-retical descriptorsrdquo Chemical Reviews vol 100 no 10pp 3827ndash3858 2000

[43] J B Liu M Younas M Habib M Yousaf and W NazeerldquoM-polynomials and degree-based topological indices ofVC5C7[p q] and HC5C7[p q] nanotubesrdquo IEEE Accessvol 7 pp 41125ndash41132 2019

[44] M Riaz W Gao and A Q Baig ldquoM-polynomials and degree-based topological indices of some families of convex poly-topesrdquo Open Journal of Mathematical Sciences vol 2 no 1pp 18ndash28 2018

[45] M S Anjum and M U Safdar ldquoK Banhatti and K hyper-Banhatti indices of nanotubesrdquo Engineering and AppliedScience Letters vol 2 no 1 pp 19ndash37 2019

[46] Z Shao A R Virk M S Javed M A Rehman andM R Farahani ldquoDegree based graph invariants for themolecular graph of Bismuth Tri-Iodiderdquo Engineering andApplied Science Letters vol 2 no 1 pp 01ndash11 2019

[47] L Yan M R Farahani and W Gao ldquoDistance-based indicescomputation of symmetry molecular structuresrdquo OpenJournal of Mathematical Sciences vol 2 no 1 pp 323ndash3372018

[48] R Kanabur and S K Hosamani ldquoSome numerical invariantsassociated with V-phenylenic nanotube and nanotorirdquo En-gineering and Applied Science Letters vol 1 no 1 pp 1ndash92018

[49] J B Liu C Wang and S Wang ldquoZagreb indices and mul-tiplicative Zagreb indices of eulerian graphsrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 42 pp 67ndash782019

[50] J B Liu and X F Pan ldquoAsymptotic Laplacian-energy-likeinvariant of latticesrdquo Applied Mathematics and Computationvol 253 pp 205ndash214 2015

[51] J B Liu and X-F Pan ldquoMinimizing Kirchhoffindex amonggraphs with a given vertex bipartitenessrdquo Applied Mathe-matics and Computation vol 291 pp 84ndash88 2016

[52] J B Liu J Zhao and Z Zhu ldquoOn the number of spanningtrees and normalized Laplacian of linear octagonal-quadri-lateral networksrdquo International Journal of Quantum Chem-istry vol 2019 Article ID e25971 2019

Journal of Chemistry 11

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Submit your manuscripts atwwwhindawicom

such that

E kk+1 ORk( 11138571113868111386811138681113868

1113868111386811138681113868 nk

E k+1k+1 ORk( 11138571113868111386811138681113868

1113868111386811138681113868 n2(k + 1) minus n(1 + 2k)

2

(50)

Now from the definition of M-polynomial we have

M(G x y) 1113944δleilejleΔ

mijxiy

j

1113944klek+1

mk(k+1)xky

k+1+ 1113944

k+1lek+1m(k+1)(k+1)x

k+1y

k+1

E kk+1 ORk( 11138571113868111386811138681113868

1113868111386811138681113868xky

k+1+ E k+1k+1 ORk( 1113857

11138681113868111386811138681113868111386811138681113868x

k+1y

k+1

nkxky

k+1+

n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(51)

Corollary 10 Let (ORk) be the swapped network then

M1 ORk( 1113857 2nk2

+ nk + n2(k + 1)

2minus n(k + 1)(1 + 2k)

(52)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(53)

en

Dx(f(x y)) nk2x

ky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

Dy(f(x y)) nk(k + 1)xky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(54)

Now we have

M1 ORk( 1113857 Dxf + Dyf11138681113868111386811138681113868xy1

2nk2

+ nk + n2(k + 1)

2minus n(k + 1)(1 + 2k)

(55)

Corollary 11 Let (ORk) be the swapped network then

M2 ORk( 1113857 nk3

+ nk2

+n2(k + 1)3

2minus

n(k + 1)2(1 + 2k)

2

(56)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(57)

en

Dy(f(x y)) nk2x

ky

k+1+ nkx

ky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxDy(f(x y)) nk3x

ky

k+1+ nk

2x

ky

k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(58)

Now we have

M2 ORk( 1113857 DxDyf11138681113868111386811138681113868xy1

nk3

+ nk2

+n2(k + 1)3

2minus

n(k + 1)2(1 + 2k)

2

(59)

Corollary 12 Let (ORk) be the swapped network then

mM2 ORk( 1113857

nk

k2 + k( )+

n2

(2k + 2)minus

n(1 + 2k)

2(k + 1)2 (60)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(61)

en

Sy(f(x y)) nk

(k + 1)1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2(k + 1)1113888 1113889x

k+1y

k+1

SxSy(f(x y)) nk

k2 + k( )1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2(k + 1)21113888 1113889x

k+1y

k+1

(62)

Journal of Chemistry 7

Now we havem

M2 ORk( 1113857 SxSyf11138681113868111386811138681113868xy1

nk

k2 + k( )+

n2

(2k + 2)minus

n(1 + 2k)

2(k + 1)2

(63)

Corollary 13 Let (ORk) be the swapped network then

Rα1113888 ORk( 1113857 nk3α

+ nk2α

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

2α1113888 1113889x

k+1y

k+11113889

(64)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(65)

en

Dαy(f(x y)) nk

2αx

ky

k+1+ nk

αx

ky

k+1

+(k + 1)α n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DαxD

αy1113888(f(x y)) nk

3αx

ky

k+1+ nk

2αx

ky

k+1

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+11113889

(66)

Now we have

Rα ORk( 1113857 DαxD

αyf

11138681113868111386811138681113868xy11113874

nk3α

+ nk2α

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

2α1113888 1113889x

k+1y

k+1

(67)

Corollary 14 Let (ORk) be the swapped network then

RRα ORk( 1113857 n

(k + 1)α+

n2(k + 1) minus n(1 + 2k)

(k + 1)2α2α1113888 11138891113888 1113889

(68)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(69)

en

Sαy(f(x y))

nkα

(k + 1)α1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2α(k + 1)α1113888 1113889x

k+1y

k+1

SαxS

αy1113888(f(x y))

n

(k + 1)α1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2α(k + 1)2α1113888 1113889x

k+1y

k+11113889

(70)

Now we have

RRα ORk( 1113857 SαxS

αyf

11138681113868111386811138681113868xy11113874

n

(k + 1)α+

n2(k + 1) minus n(1 + 2k)

(k + 1)2α2α1113888 1113889

(71)

Corollary 15 Let (ORk) be the swapped network then

SSD ORk( 1113857 nk2

(k + 1)+ nk + n + n

2(k + 1) minus n(1 + 2k)

(72)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(73)

en

SxDy(f(x y)) nkxky

k+1+ nx

ky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxSy(f(x y)) nk2

(k + 1)1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(74)

Now we have

SSD ORk( 1113857 DxSyf + SxDyf1113872 111387311138681113868111386811138681113868xy1

nk2

(k + 1)+ nk + n + n

2(k + 1) minus n(1 + 2k)

(75)

8 Journal of Chemistry

Corollary 16 Let (ORk) be the swapped network then

H ORk( 1113857 2nk

2k + 1+

n2(k + 1)

4k + 4minus

n(1 + 2k)

4k + 41113888 1113889 (76)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(77)

en

J(f(x y)) nkx2k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

2SxJ(f(x y)) 21113888nk

2k + 11113888 1113889x

2k+1

+n2(k + 1)

4k + 41113888 1113889x

2k+2

minusn(1 + 2k)

4k + 41113888 1113889x

2k+21113889

(78)

Now we have

H ORk( 1113857 2SxJf1113868111386811138681113868x1

2nk

2k + 1+

n2(k + 1)

4k + 4minus

n(1 + 2k)

4k + 41113888 1113889

(79)

Corollary 17 Let (ORk) be the swapped network then

I ORk( 1113857 nk3

2k + 1+

nk2

2k + 1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889

(80)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(81)

en

Dy(f(x y)) nk(k + 1)xky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxDy(f(x y)) nk3x

ky

k+1+ nk

2x

ky

k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

JDxDy(f(x y)) nk3x2k+1

+ nk2x2k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

SxJDxDy(f(x y)) nk3

2k + 11113888 1113889x

2k+1+

nk2

2k + 11113888 1113889x

2k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889x

2k+2

(82)

Now we have

I ORk( 1113857 SxJDxDyf11138681113868111386811138681113868x1

nk3

2k + 1+

nk2

2k + 1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889

(83)

Corollary 18 Let (ORk) be the swapped network then

A ORk( 1113857 nk5(k + 1)2

(2k minus 1)3

+nk4(k + 1)2

(2k minus 1)3

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889

(84)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(85)

Journal of Chemistry 9

en

D3y(f(x y)) nk

2(k + 1)

2x

ky

k+1+ nk(k + 1)

2x

ky

k+1

+(k + 1)3 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

D3yD

3y(f(x y)) nk

5(k + 1)

2x

ky

k+1+ nk

4(k + 1)

2x

ky

k+1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

JD3yD

3y(f(x y)) nk

5(k + 1)

2x2k+1

+ nk4(k + 1)

2x2k+1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

Qminus 2JD3yD

3y(f(x y)) nk

5(k + 1)

2x2kminus 1

+ nk4(k + 1)

2x2kminus 1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k

S3xQminus 2JD

3yD

3y(f(x y))

nk5(k + 1)2

(2k minus 1)31113888 1113889x

2kminus 1+

nk4(k + 1)2

(2k minus 1)31113888 1113889x

2kminus 1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889x2k

(86)

Now we have

A ORk( 1113857 S3xQminus 2JD

3yD

3yf

11138681113868111386811138681113868x1

nk5(k + 1)2

(2k minus 1)3+

nk4(k + 1)2

(2k minus 1)3

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889

(87)

6 Conclusion

In this paper our focus is on swapped interconnectionnetworks that allow systematic construction of large scal-able modular and robust parallel architectures whilemaintaining many desirable attributes of the underlyingbasis network comprising its clusters We have computedseveral TIs of underlined networks Firstly we computed M-polynomials of understudy networks and then we recoveredZagreb indices Randic indices and some other indices[49ndash52]

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to this study

Acknowledgments

is study was partially supported by the Natural ScienceFoundation of Anhui Province (no KJ2018A059)

References

[1] W Gao M R Farahani and L Shi ldquoForgotten topologicalindex of some drug structuresrdquo Acta Medica Mediterraneavol 32 no 1 pp 579ndash585 2016

[2] W Gao M K Siddiqui M Imran M K Jamil andM R Farahani ldquoForgotten topological index of chemicalstructure in drugsrdquo Saudi Pharmaceutical Journal vol 24no 3 pp 258ndash264 2016

[3] W Gao W Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 Article ID 3216327 8 pages 2016

[4] M Imran A Asghar and A Q Baig ldquoOn graph invariants ofoxide networkrdquo Engineering and Applied Science Lettersvol 1(2018) no 1 pp 23ndash28 2018

[5] G Liu Z Jia and W Gao ldquoOntology similarity computingbased on stochastic primal dual coordinate techniquerdquo OpenJournal of Mathematical Sciences vol 2(2018) no 1pp 221ndash227 2018

[6] S Noreen and A Mahmood ldquoZagreb polynomials andredefined Zagreb indices for the line graph of carbonnanoconesrdquo Open Journal of Mathematical Analysisvol 2(2018) no 1 pp 66ndash73 2018

[7] S Hayat and M Imran ldquoComputation of topological indicesof certain networksrdquo Applied Mathematics and Computationvol 240 pp 213ndash228 2014

[8] S Hayat and M Imran ldquoComputation of certain topologicalindices of nanotubes covered byrdquo Journal of Computationaland eoretical Nanoscience vol 12 no 4 pp 533ndash541 2015

[9] A Q Baig M Imran and H Ali ldquoOn topological indices ofpoly oxide poly silicate DOX and DSL networksrdquo CanadianJournal of Chemistry vol 93 no 7 pp 730ndash739 2015

[10] Y Bashir A Aslam M Kamran et al ldquoOn forgotten topo-logical indices of some dendrimers structurerdquo Moleculesvol 22 no 6 p 867 2017

[11] A Aslam J L Guirao S Ahmad and W Gao ldquoTopologicalindices of the line graph of subdivision graph of completebipartite graphsrdquo Applied Mathematics amp Information Sci-ences vol 11 no 6 pp 1631ndash1636 2017

[12] G C Marsden P J Marchand P Harvey and S C EsenerldquoOptical transpose interconnection system architecturesrdquoOptics Letters vol 18 no 13 pp 1083ndash1085 1993

[13] A Al-Ayyoub A Awwad K Day and M Ould-KhaoualdquoGeneralized methods for algorithm development on opticalsystemsrdquo e Journal of Supercomputing vol 38 no 2pp 111ndash125 2006

[14] C F Wang and S Sahni ldquoMatrix multiplication on the OTIS-mesh optoelectronic computerrdquo IEEE Transactions onComputers vol 50 no 7 pp 635ndash646 2001

[15] C F Wang and S Sahni ldquoImage processing on the OTIS-mesh optoelectronic computerrdquo IEEE Transactions on Paralleland Distributed Systems vol 11 no 2 pp 97ndash109 2000

[16] C F Wang and S Sahni ldquoBasic operations on the OTIS-meshoptoelectronic computerrdquo IEEE Transactions on Parallel andDistributed Systems vol 9 no 12 pp 1226ndash1236 1998

10 Journal of Chemistry

[17] S Rajasekaran and S Sahni ldquoRandomized routing selectionand sorting on the OTIS-meshrdquo IEEE Transactions on Paralleland Distributed Systems vol 9 no 9 pp 833ndash840 1998

[18] A R Virk M N Jhangeer and M A Rehman ldquoReverseZagreb and reverse hyper-zagreb indices for silicon carbideSi2C3I[rs] and Si2C3II[rs]rdquo Engineering and Applied ScienceLetters vol 1 no 2 pp 37ndash50 2018

[19] N De ldquoComputing reformulated first Zagreb index of somechemical graphs as an application of generalized hierarchicalproduct of graphsrdquo Open Journal of Mathematical Sciencesvol 2 no 1 pp 338ndash350 2018

[20] N De ldquoHyper Zagreb index of bridge and chain graphsrdquoOpen Journal of Mathematical Sciences vol 2 no 1 pp 1ndash172018

[21] A T Balaban I Motoc D Bonchev and O MekenyanldquoTopological indices for structureactivity correlationsrdquo inSteric Effects in Drug Design pp 21ndash55 Springer BerlinGermany 1983

[22] I Gutman B Ruscic N Trinajstic and C F Wilcox JrldquoGraph theory and molecular orbitals XII Acyclic polyenesrdquoe Journal of Chemical Physics vol 62 no 9 pp 3399ndash34051975

[23] W Gao M Asif and W Nazeer ldquoe study of honey combderived network via topological indicesrdquo Open Journal ofMathematical Analysis vol 2 no 2 pp 10ndash26 2018

[24] H Siddiqui and M R Farahani ldquoForgotten polynomial andforgotten index of certain interconnection networksrdquo OpenJournal of Mathematical Analysis vol 1 no 1 pp 44ndash592017

[25] W Gao B Muzaffar and W Nazeer ldquoK-Banhatti and K-hyper Banhatti indices of dominating David derived net-workrdquo Open Journal of Mathematical Analysis vol 1 no 1pp 13ndash24 2017

[26] W Gao A Asghar andW Nazeer ldquoComputing degree-basedtopological indices of jahangir graphrdquo Engineering and Ap-plied Science Letters vol 1 no 1 pp 16ndash22 2018

[27] T Doslic B Furtula A Graovac I Gutman S Moradi andZ Yarahmadi ldquoOn vertex degree based molecular structuredescriptorsrdquo MATCH Communications in Mathematical andin Computer Chemistry vol 66 no 2 pp 613ndash626 2011

[28] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975

[29] M Randic ldquoOn history of the Randic index and emerginghostility toward chemical graph theoryrdquo MATCH Commu-nications in Mathematical and in Computer Chemistryvol 59 pp 5ndash124 2008

[30] M Randic ldquoe connectivity index 25 years afterrdquo Journal ofMolecular Graphics and Modelling vol 20 no 1 pp 19ndash352001

[31] K C Das and I Gutman ldquoSome properties of the secondZagreb indexrdquo MATCH Communications in Mathematicaland in Computer Chemistry vol 52 no 1 pp 3ndash1 2004

[32] L B Kier and L H HallMolecular Connectivity in Chemistryand Drug Research Academic Press New York NY USA1976

[33] L B Kier and L H HallMolecular Connectivity in Structure-Activity Analysis Research Studies Boston MA USA 1986

[34] X Li and I Gutman Mathematical Aspects of Randic-TypeMolecular Structure Descriptors University of KragujevacKragujevac Serbia 2006

[35] B Bollobas and P Erdos ldquoGraphs of extremal weightsrdquo ArsCombinatoria vol 50 pp 225ndash233 1998

[36] Y Hu X Li Y Shi T Xu and I Gutman ldquoOn moleculargraphs with smallest and greatest zeroth-order general Randicindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 54 no 2 pp 425ndash434 2005

[37] Y Ma S Cao Y Shi I Gutman M Dehmer and B FurtulaldquoFrom the connectivity index to various Randic-type de-scriptorsrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 80 no 1 pp 85ndash106 2018

[38] I Gutman B Furtula and S B Bozkurt ldquoOn Randic energyrdquoLinear Algebra and its Applications vol 442 pp 50ndash57 2014

[39] X Li and Y Shi ldquoA survey on the Randic indexrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 59 no 1 pp 127ndash156 2008

[40] M Cavers S Fallat and S Kirkland ldquoOn the normalizedLaplacian energy and general Randic index R-1 of graphsrdquoLinear Algebra and its Applications vol 433 no 1 pp 172ndash190 2010

[41] S B Bozkurt A D Gungor I Gutman and A S CevikldquoRandic matrix and Randic energyrdquo MATCH Communica-tions in Mathematical and in Computer Chemistry vol 64pp 239ndash250 2010

[42] L Pogliani ldquoFrom molecular connectivity indices to semi-empirical connectivity terms recent trends in graph theo-retical descriptorsrdquo Chemical Reviews vol 100 no 10pp 3827ndash3858 2000

[43] J B Liu M Younas M Habib M Yousaf and W NazeerldquoM-polynomials and degree-based topological indices ofVC5C7[p q] and HC5C7[p q] nanotubesrdquo IEEE Accessvol 7 pp 41125ndash41132 2019

[44] M Riaz W Gao and A Q Baig ldquoM-polynomials and degree-based topological indices of some families of convex poly-topesrdquo Open Journal of Mathematical Sciences vol 2 no 1pp 18ndash28 2018

[45] M S Anjum and M U Safdar ldquoK Banhatti and K hyper-Banhatti indices of nanotubesrdquo Engineering and AppliedScience Letters vol 2 no 1 pp 19ndash37 2019

[46] Z Shao A R Virk M S Javed M A Rehman andM R Farahani ldquoDegree based graph invariants for themolecular graph of Bismuth Tri-Iodiderdquo Engineering andApplied Science Letters vol 2 no 1 pp 01ndash11 2019

[47] L Yan M R Farahani and W Gao ldquoDistance-based indicescomputation of symmetry molecular structuresrdquo OpenJournal of Mathematical Sciences vol 2 no 1 pp 323ndash3372018

[48] R Kanabur and S K Hosamani ldquoSome numerical invariantsassociated with V-phenylenic nanotube and nanotorirdquo En-gineering and Applied Science Letters vol 1 no 1 pp 1ndash92018

[49] J B Liu C Wang and S Wang ldquoZagreb indices and mul-tiplicative Zagreb indices of eulerian graphsrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 42 pp 67ndash782019

[50] J B Liu and X F Pan ldquoAsymptotic Laplacian-energy-likeinvariant of latticesrdquo Applied Mathematics and Computationvol 253 pp 205ndash214 2015

[51] J B Liu and X-F Pan ldquoMinimizing Kirchhoffindex amonggraphs with a given vertex bipartitenessrdquo Applied Mathe-matics and Computation vol 291 pp 84ndash88 2016

[52] J B Liu J Zhao and Z Zhu ldquoOn the number of spanningtrees and normalized Laplacian of linear octagonal-quadri-lateral networksrdquo International Journal of Quantum Chem-istry vol 2019 Article ID e25971 2019

Journal of Chemistry 11

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2018

Bioinorganic Chemistry and ApplicationsHindawiwwwhindawicom Volume 2018

SpectroscopyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Medicinal ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Biochemistry Research International

Hindawiwwwhindawicom Volume 2018

Enzyme Research

Hindawiwwwhindawicom Volume 2018

Journal of

SpectroscopyAnalytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

MaterialsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

BioMed Research International Electrochemistry

International Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

Now we havem

M2 ORk( 1113857 SxSyf11138681113868111386811138681113868xy1

nk

k2 + k( )+

n2

(2k + 2)minus

n(1 + 2k)

2(k + 1)2

(63)

Corollary 13 Let (ORk) be the swapped network then

Rα1113888 ORk( 1113857 nk3α

+ nk2α

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

2α1113888 1113889x

k+1y

k+11113889

(64)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(65)

en

Dαy(f(x y)) nk

2αx

ky

k+1+ nk

αx

ky

k+1

+(k + 1)α n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DαxD

αy1113888(f(x y)) nk

3αx

ky

k+1+ nk

2αx

ky

k+1

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+11113889

(66)

Now we have

Rα ORk( 1113857 DαxD

αyf

11138681113868111386811138681113868xy11113874

nk3α

+ nk2α

+(k + 1)2α n2(k + 1) minus n(1 + 2k)

2α1113888 1113889x

k+1y

k+1

(67)

Corollary 14 Let (ORk) be the swapped network then

RRα ORk( 1113857 n

(k + 1)α+

n2(k + 1) minus n(1 + 2k)

(k + 1)2α2α1113888 11138891113888 1113889

(68)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(69)

en

Sαy(f(x y))

nkα

(k + 1)α1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2α(k + 1)α1113888 1113889x

k+1y

k+1

SαxS

αy1113888(f(x y))

n

(k + 1)α1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

2α(k + 1)2α1113888 1113889x

k+1y

k+11113889

(70)

Now we have

RRα ORk( 1113857 SαxS

αyf

11138681113868111386811138681113868xy11113874

n

(k + 1)α+

n2(k + 1) minus n(1 + 2k)

(k + 1)2α2α1113888 1113889

(71)

Corollary 15 Let (ORk) be the swapped network then

SSD ORk( 1113857 nk2

(k + 1)+ nk + n + n

2(k + 1) minus n(1 + 2k)

(72)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(73)

en

SxDy(f(x y)) nkxky

k+1+ nx

ky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxSy(f(x y)) nk2

(k + 1)1113888 1113889x

ky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(74)

Now we have

SSD ORk( 1113857 DxSyf + SxDyf1113872 111387311138681113868111386811138681113868xy1

nk2

(k + 1)+ nk + n + n

2(k + 1) minus n(1 + 2k)

(75)

8 Journal of Chemistry

Corollary 16 Let (ORk) be the swapped network then

H ORk( 1113857 2nk

2k + 1+

n2(k + 1)

4k + 4minus

n(1 + 2k)

4k + 41113888 1113889 (76)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(77)

en

J(f(x y)) nkx2k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

2SxJ(f(x y)) 21113888nk

2k + 11113888 1113889x

2k+1

+n2(k + 1)

4k + 41113888 1113889x

2k+2

minusn(1 + 2k)

4k + 41113888 1113889x

2k+21113889

(78)

Now we have

H ORk( 1113857 2SxJf1113868111386811138681113868x1

2nk

2k + 1+

n2(k + 1)

4k + 4minus

n(1 + 2k)

4k + 41113888 1113889

(79)

Corollary 17 Let (ORk) be the swapped network then

I ORk( 1113857 nk3

2k + 1+

nk2

2k + 1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889

(80)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(81)

en

Dy(f(x y)) nk(k + 1)xky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxDy(f(x y)) nk3x

ky

k+1+ nk

2x

ky

k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

JDxDy(f(x y)) nk3x2k+1

+ nk2x2k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

SxJDxDy(f(x y)) nk3

2k + 11113888 1113889x

2k+1+

nk2

2k + 11113888 1113889x

2k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889x

2k+2

(82)

Now we have

I ORk( 1113857 SxJDxDyf11138681113868111386811138681113868x1

nk3

2k + 1+

nk2

2k + 1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889

(83)

Corollary 18 Let (ORk) be the swapped network then

A ORk( 1113857 nk5(k + 1)2

(2k minus 1)3

+nk4(k + 1)2

(2k minus 1)3

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889

(84)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(85)

Journal of Chemistry 9

en

D3y(f(x y)) nk

2(k + 1)

2x

ky

k+1+ nk(k + 1)

2x

ky

k+1

+(k + 1)3 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

D3yD

3y(f(x y)) nk

5(k + 1)

2x

ky

k+1+ nk

4(k + 1)

2x

ky

k+1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

JD3yD

3y(f(x y)) nk

5(k + 1)

2x2k+1

+ nk4(k + 1)

2x2k+1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

Qminus 2JD3yD

3y(f(x y)) nk

5(k + 1)

2x2kminus 1

+ nk4(k + 1)

2x2kminus 1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k

S3xQminus 2JD

3yD

3y(f(x y))

nk5(k + 1)2

(2k minus 1)31113888 1113889x

2kminus 1+

nk4(k + 1)2

(2k minus 1)31113888 1113889x

2kminus 1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889x2k

(86)

Now we have

A ORk( 1113857 S3xQminus 2JD

3yD

3yf

11138681113868111386811138681113868x1

nk5(k + 1)2

(2k minus 1)3+

nk4(k + 1)2

(2k minus 1)3

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889

(87)

6 Conclusion

In this paper our focus is on swapped interconnectionnetworks that allow systematic construction of large scal-able modular and robust parallel architectures whilemaintaining many desirable attributes of the underlyingbasis network comprising its clusters We have computedseveral TIs of underlined networks Firstly we computed M-polynomials of understudy networks and then we recoveredZagreb indices Randic indices and some other indices[49ndash52]

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to this study

Acknowledgments

is study was partially supported by the Natural ScienceFoundation of Anhui Province (no KJ2018A059)

References

[1] W Gao M R Farahani and L Shi ldquoForgotten topologicalindex of some drug structuresrdquo Acta Medica Mediterraneavol 32 no 1 pp 579ndash585 2016

[2] W Gao M K Siddiqui M Imran M K Jamil andM R Farahani ldquoForgotten topological index of chemicalstructure in drugsrdquo Saudi Pharmaceutical Journal vol 24no 3 pp 258ndash264 2016

[3] W Gao W Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 Article ID 3216327 8 pages 2016

[4] M Imran A Asghar and A Q Baig ldquoOn graph invariants ofoxide networkrdquo Engineering and Applied Science Lettersvol 1(2018) no 1 pp 23ndash28 2018

[5] G Liu Z Jia and W Gao ldquoOntology similarity computingbased on stochastic primal dual coordinate techniquerdquo OpenJournal of Mathematical Sciences vol 2(2018) no 1pp 221ndash227 2018

[6] S Noreen and A Mahmood ldquoZagreb polynomials andredefined Zagreb indices for the line graph of carbonnanoconesrdquo Open Journal of Mathematical Analysisvol 2(2018) no 1 pp 66ndash73 2018

[7] S Hayat and M Imran ldquoComputation of topological indicesof certain networksrdquo Applied Mathematics and Computationvol 240 pp 213ndash228 2014

[8] S Hayat and M Imran ldquoComputation of certain topologicalindices of nanotubes covered byrdquo Journal of Computationaland eoretical Nanoscience vol 12 no 4 pp 533ndash541 2015

[9] A Q Baig M Imran and H Ali ldquoOn topological indices ofpoly oxide poly silicate DOX and DSL networksrdquo CanadianJournal of Chemistry vol 93 no 7 pp 730ndash739 2015

[10] Y Bashir A Aslam M Kamran et al ldquoOn forgotten topo-logical indices of some dendrimers structurerdquo Moleculesvol 22 no 6 p 867 2017

[11] A Aslam J L Guirao S Ahmad and W Gao ldquoTopologicalindices of the line graph of subdivision graph of completebipartite graphsrdquo Applied Mathematics amp Information Sci-ences vol 11 no 6 pp 1631ndash1636 2017

[12] G C Marsden P J Marchand P Harvey and S C EsenerldquoOptical transpose interconnection system architecturesrdquoOptics Letters vol 18 no 13 pp 1083ndash1085 1993

[13] A Al-Ayyoub A Awwad K Day and M Ould-KhaoualdquoGeneralized methods for algorithm development on opticalsystemsrdquo e Journal of Supercomputing vol 38 no 2pp 111ndash125 2006

[14] C F Wang and S Sahni ldquoMatrix multiplication on the OTIS-mesh optoelectronic computerrdquo IEEE Transactions onComputers vol 50 no 7 pp 635ndash646 2001

[15] C F Wang and S Sahni ldquoImage processing on the OTIS-mesh optoelectronic computerrdquo IEEE Transactions on Paralleland Distributed Systems vol 11 no 2 pp 97ndash109 2000

[16] C F Wang and S Sahni ldquoBasic operations on the OTIS-meshoptoelectronic computerrdquo IEEE Transactions on Parallel andDistributed Systems vol 9 no 12 pp 1226ndash1236 1998

10 Journal of Chemistry

[17] S Rajasekaran and S Sahni ldquoRandomized routing selectionand sorting on the OTIS-meshrdquo IEEE Transactions on Paralleland Distributed Systems vol 9 no 9 pp 833ndash840 1998

[18] A R Virk M N Jhangeer and M A Rehman ldquoReverseZagreb and reverse hyper-zagreb indices for silicon carbideSi2C3I[rs] and Si2C3II[rs]rdquo Engineering and Applied ScienceLetters vol 1 no 2 pp 37ndash50 2018

[19] N De ldquoComputing reformulated first Zagreb index of somechemical graphs as an application of generalized hierarchicalproduct of graphsrdquo Open Journal of Mathematical Sciencesvol 2 no 1 pp 338ndash350 2018

[20] N De ldquoHyper Zagreb index of bridge and chain graphsrdquoOpen Journal of Mathematical Sciences vol 2 no 1 pp 1ndash172018

[21] A T Balaban I Motoc D Bonchev and O MekenyanldquoTopological indices for structureactivity correlationsrdquo inSteric Effects in Drug Design pp 21ndash55 Springer BerlinGermany 1983

[22] I Gutman B Ruscic N Trinajstic and C F Wilcox JrldquoGraph theory and molecular orbitals XII Acyclic polyenesrdquoe Journal of Chemical Physics vol 62 no 9 pp 3399ndash34051975

[23] W Gao M Asif and W Nazeer ldquoe study of honey combderived network via topological indicesrdquo Open Journal ofMathematical Analysis vol 2 no 2 pp 10ndash26 2018

[24] H Siddiqui and M R Farahani ldquoForgotten polynomial andforgotten index of certain interconnection networksrdquo OpenJournal of Mathematical Analysis vol 1 no 1 pp 44ndash592017

[25] W Gao B Muzaffar and W Nazeer ldquoK-Banhatti and K-hyper Banhatti indices of dominating David derived net-workrdquo Open Journal of Mathematical Analysis vol 1 no 1pp 13ndash24 2017

[26] W Gao A Asghar andW Nazeer ldquoComputing degree-basedtopological indices of jahangir graphrdquo Engineering and Ap-plied Science Letters vol 1 no 1 pp 16ndash22 2018

[27] T Doslic B Furtula A Graovac I Gutman S Moradi andZ Yarahmadi ldquoOn vertex degree based molecular structuredescriptorsrdquo MATCH Communications in Mathematical andin Computer Chemistry vol 66 no 2 pp 613ndash626 2011

[28] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975

[29] M Randic ldquoOn history of the Randic index and emerginghostility toward chemical graph theoryrdquo MATCH Commu-nications in Mathematical and in Computer Chemistryvol 59 pp 5ndash124 2008

[30] M Randic ldquoe connectivity index 25 years afterrdquo Journal ofMolecular Graphics and Modelling vol 20 no 1 pp 19ndash352001

[31] K C Das and I Gutman ldquoSome properties of the secondZagreb indexrdquo MATCH Communications in Mathematicaland in Computer Chemistry vol 52 no 1 pp 3ndash1 2004

[32] L B Kier and L H HallMolecular Connectivity in Chemistryand Drug Research Academic Press New York NY USA1976

[33] L B Kier and L H HallMolecular Connectivity in Structure-Activity Analysis Research Studies Boston MA USA 1986

[34] X Li and I Gutman Mathematical Aspects of Randic-TypeMolecular Structure Descriptors University of KragujevacKragujevac Serbia 2006

[35] B Bollobas and P Erdos ldquoGraphs of extremal weightsrdquo ArsCombinatoria vol 50 pp 225ndash233 1998

[36] Y Hu X Li Y Shi T Xu and I Gutman ldquoOn moleculargraphs with smallest and greatest zeroth-order general Randicindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 54 no 2 pp 425ndash434 2005

[37] Y Ma S Cao Y Shi I Gutman M Dehmer and B FurtulaldquoFrom the connectivity index to various Randic-type de-scriptorsrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 80 no 1 pp 85ndash106 2018

[38] I Gutman B Furtula and S B Bozkurt ldquoOn Randic energyrdquoLinear Algebra and its Applications vol 442 pp 50ndash57 2014

[39] X Li and Y Shi ldquoA survey on the Randic indexrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 59 no 1 pp 127ndash156 2008

[40] M Cavers S Fallat and S Kirkland ldquoOn the normalizedLaplacian energy and general Randic index R-1 of graphsrdquoLinear Algebra and its Applications vol 433 no 1 pp 172ndash190 2010

[41] S B Bozkurt A D Gungor I Gutman and A S CevikldquoRandic matrix and Randic energyrdquo MATCH Communica-tions in Mathematical and in Computer Chemistry vol 64pp 239ndash250 2010

[42] L Pogliani ldquoFrom molecular connectivity indices to semi-empirical connectivity terms recent trends in graph theo-retical descriptorsrdquo Chemical Reviews vol 100 no 10pp 3827ndash3858 2000

[43] J B Liu M Younas M Habib M Yousaf and W NazeerldquoM-polynomials and degree-based topological indices ofVC5C7[p q] and HC5C7[p q] nanotubesrdquo IEEE Accessvol 7 pp 41125ndash41132 2019

[44] M Riaz W Gao and A Q Baig ldquoM-polynomials and degree-based topological indices of some families of convex poly-topesrdquo Open Journal of Mathematical Sciences vol 2 no 1pp 18ndash28 2018

[45] M S Anjum and M U Safdar ldquoK Banhatti and K hyper-Banhatti indices of nanotubesrdquo Engineering and AppliedScience Letters vol 2 no 1 pp 19ndash37 2019

[46] Z Shao A R Virk M S Javed M A Rehman andM R Farahani ldquoDegree based graph invariants for themolecular graph of Bismuth Tri-Iodiderdquo Engineering andApplied Science Letters vol 2 no 1 pp 01ndash11 2019

[47] L Yan M R Farahani and W Gao ldquoDistance-based indicescomputation of symmetry molecular structuresrdquo OpenJournal of Mathematical Sciences vol 2 no 1 pp 323ndash3372018

[48] R Kanabur and S K Hosamani ldquoSome numerical invariantsassociated with V-phenylenic nanotube and nanotorirdquo En-gineering and Applied Science Letters vol 1 no 1 pp 1ndash92018

[49] J B Liu C Wang and S Wang ldquoZagreb indices and mul-tiplicative Zagreb indices of eulerian graphsrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 42 pp 67ndash782019

[50] J B Liu and X F Pan ldquoAsymptotic Laplacian-energy-likeinvariant of latticesrdquo Applied Mathematics and Computationvol 253 pp 205ndash214 2015

[51] J B Liu and X-F Pan ldquoMinimizing Kirchhoffindex amonggraphs with a given vertex bipartitenessrdquo Applied Mathe-matics and Computation vol 291 pp 84ndash88 2016

[52] J B Liu J Zhao and Z Zhu ldquoOn the number of spanningtrees and normalized Laplacian of linear octagonal-quadri-lateral networksrdquo International Journal of Quantum Chem-istry vol 2019 Article ID e25971 2019

Journal of Chemistry 11

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2018

Bioinorganic Chemistry and ApplicationsHindawiwwwhindawicom Volume 2018

SpectroscopyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Medicinal ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Biochemistry Research International

Hindawiwwwhindawicom Volume 2018

Enzyme Research

Hindawiwwwhindawicom Volume 2018

Journal of

SpectroscopyAnalytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

MaterialsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

BioMed Research International Electrochemistry

International Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

Corollary 16 Let (ORk) be the swapped network then

H ORk( 1113857 2nk

2k + 1+

n2(k + 1)

4k + 4minus

n(1 + 2k)

4k + 41113888 1113889 (76)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(77)

en

J(f(x y)) nkx2k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

2SxJ(f(x y)) 21113888nk

2k + 11113888 1113889x

2k+1

+n2(k + 1)

4k + 41113888 1113889x

2k+2

minusn(1 + 2k)

4k + 41113888 1113889x

2k+21113889

(78)

Now we have

H ORk( 1113857 2SxJf1113868111386811138681113868x1

2nk

2k + 1+

n2(k + 1)

4k + 4minus

n(1 + 2k)

4k + 41113888 1113889

(79)

Corollary 17 Let (ORk) be the swapped network then

I ORk( 1113857 nk3

2k + 1+

nk2

2k + 1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889

(80)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(81)

en

Dy(f(x y)) nk(k + 1)xky

k+1

+(k + 1)n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

DxDy(f(x y)) nk3x

ky

k+1+ nk

2x

ky

k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

JDxDy(f(x y)) nk3x2k+1

+ nk2x2k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

SxJDxDy(f(x y)) nk3

2k + 11113888 1113889x

2k+1+

nk2

2k + 11113888 1113889x

2k+1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889x

2k+2

(82)

Now we have

I ORk( 1113857 SxJDxDyf11138681113868111386811138681113868x1

nk3

2k + 1+

nk2

2k + 1

+(k + 1)2 n2(k + 1) minus n(1 + 2k)

4k + 41113888 1113889

(83)

Corollary 18 Let (ORk) be the swapped network then

A ORk( 1113857 nk5(k + 1)2

(2k minus 1)3

+nk4(k + 1)2

(2k minus 1)3

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889

(84)

Proof Let

f(x y) M(G x y) nkxky

k+1

+n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

(85)

Journal of Chemistry 9

en

D3y(f(x y)) nk

2(k + 1)

2x

ky

k+1+ nk(k + 1)

2x

ky

k+1

+(k + 1)3 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

D3yD

3y(f(x y)) nk

5(k + 1)

2x

ky

k+1+ nk

4(k + 1)

2x

ky

k+1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

JD3yD

3y(f(x y)) nk

5(k + 1)

2x2k+1

+ nk4(k + 1)

2x2k+1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

Qminus 2JD3yD

3y(f(x y)) nk

5(k + 1)

2x2kminus 1

+ nk4(k + 1)

2x2kminus 1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k

S3xQminus 2JD

3yD

3y(f(x y))

nk5(k + 1)2

(2k minus 1)31113888 1113889x

2kminus 1+

nk4(k + 1)2

(2k minus 1)31113888 1113889x

2kminus 1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889x2k

(86)

Now we have

A ORk( 1113857 S3xQminus 2JD

3yD

3yf

11138681113868111386811138681113868x1

nk5(k + 1)2

(2k minus 1)3+

nk4(k + 1)2

(2k minus 1)3

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889

(87)

6 Conclusion

In this paper our focus is on swapped interconnectionnetworks that allow systematic construction of large scal-able modular and robust parallel architectures whilemaintaining many desirable attributes of the underlyingbasis network comprising its clusters We have computedseveral TIs of underlined networks Firstly we computed M-polynomials of understudy networks and then we recoveredZagreb indices Randic indices and some other indices[49ndash52]

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to this study

Acknowledgments

is study was partially supported by the Natural ScienceFoundation of Anhui Province (no KJ2018A059)

References

[1] W Gao M R Farahani and L Shi ldquoForgotten topologicalindex of some drug structuresrdquo Acta Medica Mediterraneavol 32 no 1 pp 579ndash585 2016

[2] W Gao M K Siddiqui M Imran M K Jamil andM R Farahani ldquoForgotten topological index of chemicalstructure in drugsrdquo Saudi Pharmaceutical Journal vol 24no 3 pp 258ndash264 2016

[3] W Gao W Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 Article ID 3216327 8 pages 2016

[4] M Imran A Asghar and A Q Baig ldquoOn graph invariants ofoxide networkrdquo Engineering and Applied Science Lettersvol 1(2018) no 1 pp 23ndash28 2018

[5] G Liu Z Jia and W Gao ldquoOntology similarity computingbased on stochastic primal dual coordinate techniquerdquo OpenJournal of Mathematical Sciences vol 2(2018) no 1pp 221ndash227 2018

[6] S Noreen and A Mahmood ldquoZagreb polynomials andredefined Zagreb indices for the line graph of carbonnanoconesrdquo Open Journal of Mathematical Analysisvol 2(2018) no 1 pp 66ndash73 2018

[7] S Hayat and M Imran ldquoComputation of topological indicesof certain networksrdquo Applied Mathematics and Computationvol 240 pp 213ndash228 2014

[8] S Hayat and M Imran ldquoComputation of certain topologicalindices of nanotubes covered byrdquo Journal of Computationaland eoretical Nanoscience vol 12 no 4 pp 533ndash541 2015

[9] A Q Baig M Imran and H Ali ldquoOn topological indices ofpoly oxide poly silicate DOX and DSL networksrdquo CanadianJournal of Chemistry vol 93 no 7 pp 730ndash739 2015

[10] Y Bashir A Aslam M Kamran et al ldquoOn forgotten topo-logical indices of some dendrimers structurerdquo Moleculesvol 22 no 6 p 867 2017

[11] A Aslam J L Guirao S Ahmad and W Gao ldquoTopologicalindices of the line graph of subdivision graph of completebipartite graphsrdquo Applied Mathematics amp Information Sci-ences vol 11 no 6 pp 1631ndash1636 2017

[12] G C Marsden P J Marchand P Harvey and S C EsenerldquoOptical transpose interconnection system architecturesrdquoOptics Letters vol 18 no 13 pp 1083ndash1085 1993

[13] A Al-Ayyoub A Awwad K Day and M Ould-KhaoualdquoGeneralized methods for algorithm development on opticalsystemsrdquo e Journal of Supercomputing vol 38 no 2pp 111ndash125 2006

[14] C F Wang and S Sahni ldquoMatrix multiplication on the OTIS-mesh optoelectronic computerrdquo IEEE Transactions onComputers vol 50 no 7 pp 635ndash646 2001

[15] C F Wang and S Sahni ldquoImage processing on the OTIS-mesh optoelectronic computerrdquo IEEE Transactions on Paralleland Distributed Systems vol 11 no 2 pp 97ndash109 2000

[16] C F Wang and S Sahni ldquoBasic operations on the OTIS-meshoptoelectronic computerrdquo IEEE Transactions on Parallel andDistributed Systems vol 9 no 12 pp 1226ndash1236 1998

10 Journal of Chemistry

[17] S Rajasekaran and S Sahni ldquoRandomized routing selectionand sorting on the OTIS-meshrdquo IEEE Transactions on Paralleland Distributed Systems vol 9 no 9 pp 833ndash840 1998

[18] A R Virk M N Jhangeer and M A Rehman ldquoReverseZagreb and reverse hyper-zagreb indices for silicon carbideSi2C3I[rs] and Si2C3II[rs]rdquo Engineering and Applied ScienceLetters vol 1 no 2 pp 37ndash50 2018

[19] N De ldquoComputing reformulated first Zagreb index of somechemical graphs as an application of generalized hierarchicalproduct of graphsrdquo Open Journal of Mathematical Sciencesvol 2 no 1 pp 338ndash350 2018

[20] N De ldquoHyper Zagreb index of bridge and chain graphsrdquoOpen Journal of Mathematical Sciences vol 2 no 1 pp 1ndash172018

[21] A T Balaban I Motoc D Bonchev and O MekenyanldquoTopological indices for structureactivity correlationsrdquo inSteric Effects in Drug Design pp 21ndash55 Springer BerlinGermany 1983

[22] I Gutman B Ruscic N Trinajstic and C F Wilcox JrldquoGraph theory and molecular orbitals XII Acyclic polyenesrdquoe Journal of Chemical Physics vol 62 no 9 pp 3399ndash34051975

[23] W Gao M Asif and W Nazeer ldquoe study of honey combderived network via topological indicesrdquo Open Journal ofMathematical Analysis vol 2 no 2 pp 10ndash26 2018

[24] H Siddiqui and M R Farahani ldquoForgotten polynomial andforgotten index of certain interconnection networksrdquo OpenJournal of Mathematical Analysis vol 1 no 1 pp 44ndash592017

[25] W Gao B Muzaffar and W Nazeer ldquoK-Banhatti and K-hyper Banhatti indices of dominating David derived net-workrdquo Open Journal of Mathematical Analysis vol 1 no 1pp 13ndash24 2017

[26] W Gao A Asghar andW Nazeer ldquoComputing degree-basedtopological indices of jahangir graphrdquo Engineering and Ap-plied Science Letters vol 1 no 1 pp 16ndash22 2018

[27] T Doslic B Furtula A Graovac I Gutman S Moradi andZ Yarahmadi ldquoOn vertex degree based molecular structuredescriptorsrdquo MATCH Communications in Mathematical andin Computer Chemistry vol 66 no 2 pp 613ndash626 2011

[28] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975

[29] M Randic ldquoOn history of the Randic index and emerginghostility toward chemical graph theoryrdquo MATCH Commu-nications in Mathematical and in Computer Chemistryvol 59 pp 5ndash124 2008

[30] M Randic ldquoe connectivity index 25 years afterrdquo Journal ofMolecular Graphics and Modelling vol 20 no 1 pp 19ndash352001

[31] K C Das and I Gutman ldquoSome properties of the secondZagreb indexrdquo MATCH Communications in Mathematicaland in Computer Chemistry vol 52 no 1 pp 3ndash1 2004

[32] L B Kier and L H HallMolecular Connectivity in Chemistryand Drug Research Academic Press New York NY USA1976

[33] L B Kier and L H HallMolecular Connectivity in Structure-Activity Analysis Research Studies Boston MA USA 1986

[34] X Li and I Gutman Mathematical Aspects of Randic-TypeMolecular Structure Descriptors University of KragujevacKragujevac Serbia 2006

[35] B Bollobas and P Erdos ldquoGraphs of extremal weightsrdquo ArsCombinatoria vol 50 pp 225ndash233 1998

[36] Y Hu X Li Y Shi T Xu and I Gutman ldquoOn moleculargraphs with smallest and greatest zeroth-order general Randicindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 54 no 2 pp 425ndash434 2005

[37] Y Ma S Cao Y Shi I Gutman M Dehmer and B FurtulaldquoFrom the connectivity index to various Randic-type de-scriptorsrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 80 no 1 pp 85ndash106 2018

[38] I Gutman B Furtula and S B Bozkurt ldquoOn Randic energyrdquoLinear Algebra and its Applications vol 442 pp 50ndash57 2014

[39] X Li and Y Shi ldquoA survey on the Randic indexrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 59 no 1 pp 127ndash156 2008

[40] M Cavers S Fallat and S Kirkland ldquoOn the normalizedLaplacian energy and general Randic index R-1 of graphsrdquoLinear Algebra and its Applications vol 433 no 1 pp 172ndash190 2010

[41] S B Bozkurt A D Gungor I Gutman and A S CevikldquoRandic matrix and Randic energyrdquo MATCH Communica-tions in Mathematical and in Computer Chemistry vol 64pp 239ndash250 2010

[42] L Pogliani ldquoFrom molecular connectivity indices to semi-empirical connectivity terms recent trends in graph theo-retical descriptorsrdquo Chemical Reviews vol 100 no 10pp 3827ndash3858 2000

[43] J B Liu M Younas M Habib M Yousaf and W NazeerldquoM-polynomials and degree-based topological indices ofVC5C7[p q] and HC5C7[p q] nanotubesrdquo IEEE Accessvol 7 pp 41125ndash41132 2019

[44] M Riaz W Gao and A Q Baig ldquoM-polynomials and degree-based topological indices of some families of convex poly-topesrdquo Open Journal of Mathematical Sciences vol 2 no 1pp 18ndash28 2018

[45] M S Anjum and M U Safdar ldquoK Banhatti and K hyper-Banhatti indices of nanotubesrdquo Engineering and AppliedScience Letters vol 2 no 1 pp 19ndash37 2019

[46] Z Shao A R Virk M S Javed M A Rehman andM R Farahani ldquoDegree based graph invariants for themolecular graph of Bismuth Tri-Iodiderdquo Engineering andApplied Science Letters vol 2 no 1 pp 01ndash11 2019

[47] L Yan M R Farahani and W Gao ldquoDistance-based indicescomputation of symmetry molecular structuresrdquo OpenJournal of Mathematical Sciences vol 2 no 1 pp 323ndash3372018

[48] R Kanabur and S K Hosamani ldquoSome numerical invariantsassociated with V-phenylenic nanotube and nanotorirdquo En-gineering and Applied Science Letters vol 1 no 1 pp 1ndash92018

[49] J B Liu C Wang and S Wang ldquoZagreb indices and mul-tiplicative Zagreb indices of eulerian graphsrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 42 pp 67ndash782019

[50] J B Liu and X F Pan ldquoAsymptotic Laplacian-energy-likeinvariant of latticesrdquo Applied Mathematics and Computationvol 253 pp 205ndash214 2015

[51] J B Liu and X-F Pan ldquoMinimizing Kirchhoffindex amonggraphs with a given vertex bipartitenessrdquo Applied Mathe-matics and Computation vol 291 pp 84ndash88 2016

[52] J B Liu J Zhao and Z Zhu ldquoOn the number of spanningtrees and normalized Laplacian of linear octagonal-quadri-lateral networksrdquo International Journal of Quantum Chem-istry vol 2019 Article ID e25971 2019

Journal of Chemistry 11

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2018

Bioinorganic Chemistry and ApplicationsHindawiwwwhindawicom Volume 2018

SpectroscopyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Medicinal ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Biochemistry Research International

Hindawiwwwhindawicom Volume 2018

Enzyme Research

Hindawiwwwhindawicom Volume 2018

Journal of

SpectroscopyAnalytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

MaterialsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

BioMed Research International Electrochemistry

International Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

en

D3y(f(x y)) nk

2(k + 1)

2x

ky

k+1+ nk(k + 1)

2x

ky

k+1

+(k + 1)3 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

D3yD

3y(f(x y)) nk

5(k + 1)

2x

ky

k+1+ nk

4(k + 1)

2x

ky

k+1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

k+1y

k+1

JD3yD

3y(f(x y)) nk

5(k + 1)

2x2k+1

+ nk4(k + 1)

2x2k+1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k+2

Qminus 2JD3yD

3y(f(x y)) nk

5(k + 1)

2x2kminus 1

+ nk4(k + 1)

2x2kminus 1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

21113888 1113889x

2k

S3xQminus 2JD

3yD

3y(f(x y))

nk5(k + 1)2

(2k minus 1)31113888 1113889x

2kminus 1+

nk4(k + 1)2

(2k minus 1)31113888 1113889x

2kminus 1

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889x2k

(86)

Now we have

A ORk( 1113857 S3xQminus 2JD

3yD

3yf

11138681113868111386811138681113868x1

nk5(k + 1)2

(2k minus 1)3+

nk4(k + 1)2

(2k minus 1)3

+(k + 1)6 n2(k + 1) minus n(1 + 2k)

8k31113888 1113889

(87)

6 Conclusion

In this paper our focus is on swapped interconnectionnetworks that allow systematic construction of large scal-able modular and robust parallel architectures whilemaintaining many desirable attributes of the underlyingbasis network comprising its clusters We have computedseveral TIs of underlined networks Firstly we computed M-polynomials of understudy networks and then we recoveredZagreb indices Randic indices and some other indices[49ndash52]

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to this study

Acknowledgments

is study was partially supported by the Natural ScienceFoundation of Anhui Province (no KJ2018A059)

References

[1] W Gao M R Farahani and L Shi ldquoForgotten topologicalindex of some drug structuresrdquo Acta Medica Mediterraneavol 32 no 1 pp 579ndash585 2016

[2] W Gao M K Siddiqui M Imran M K Jamil andM R Farahani ldquoForgotten topological index of chemicalstructure in drugsrdquo Saudi Pharmaceutical Journal vol 24no 3 pp 258ndash264 2016

[3] W Gao W Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 Article ID 3216327 8 pages 2016

[4] M Imran A Asghar and A Q Baig ldquoOn graph invariants ofoxide networkrdquo Engineering and Applied Science Lettersvol 1(2018) no 1 pp 23ndash28 2018

[5] G Liu Z Jia and W Gao ldquoOntology similarity computingbased on stochastic primal dual coordinate techniquerdquo OpenJournal of Mathematical Sciences vol 2(2018) no 1pp 221ndash227 2018

[6] S Noreen and A Mahmood ldquoZagreb polynomials andredefined Zagreb indices for the line graph of carbonnanoconesrdquo Open Journal of Mathematical Analysisvol 2(2018) no 1 pp 66ndash73 2018

[7] S Hayat and M Imran ldquoComputation of topological indicesof certain networksrdquo Applied Mathematics and Computationvol 240 pp 213ndash228 2014

[8] S Hayat and M Imran ldquoComputation of certain topologicalindices of nanotubes covered byrdquo Journal of Computationaland eoretical Nanoscience vol 12 no 4 pp 533ndash541 2015

[9] A Q Baig M Imran and H Ali ldquoOn topological indices ofpoly oxide poly silicate DOX and DSL networksrdquo CanadianJournal of Chemistry vol 93 no 7 pp 730ndash739 2015

[10] Y Bashir A Aslam M Kamran et al ldquoOn forgotten topo-logical indices of some dendrimers structurerdquo Moleculesvol 22 no 6 p 867 2017

[11] A Aslam J L Guirao S Ahmad and W Gao ldquoTopologicalindices of the line graph of subdivision graph of completebipartite graphsrdquo Applied Mathematics amp Information Sci-ences vol 11 no 6 pp 1631ndash1636 2017

[12] G C Marsden P J Marchand P Harvey and S C EsenerldquoOptical transpose interconnection system architecturesrdquoOptics Letters vol 18 no 13 pp 1083ndash1085 1993

[13] A Al-Ayyoub A Awwad K Day and M Ould-KhaoualdquoGeneralized methods for algorithm development on opticalsystemsrdquo e Journal of Supercomputing vol 38 no 2pp 111ndash125 2006

[14] C F Wang and S Sahni ldquoMatrix multiplication on the OTIS-mesh optoelectronic computerrdquo IEEE Transactions onComputers vol 50 no 7 pp 635ndash646 2001

[15] C F Wang and S Sahni ldquoImage processing on the OTIS-mesh optoelectronic computerrdquo IEEE Transactions on Paralleland Distributed Systems vol 11 no 2 pp 97ndash109 2000

[16] C F Wang and S Sahni ldquoBasic operations on the OTIS-meshoptoelectronic computerrdquo IEEE Transactions on Parallel andDistributed Systems vol 9 no 12 pp 1226ndash1236 1998

10 Journal of Chemistry

[17] S Rajasekaran and S Sahni ldquoRandomized routing selectionand sorting on the OTIS-meshrdquo IEEE Transactions on Paralleland Distributed Systems vol 9 no 9 pp 833ndash840 1998

[18] A R Virk M N Jhangeer and M A Rehman ldquoReverseZagreb and reverse hyper-zagreb indices for silicon carbideSi2C3I[rs] and Si2C3II[rs]rdquo Engineering and Applied ScienceLetters vol 1 no 2 pp 37ndash50 2018

[19] N De ldquoComputing reformulated first Zagreb index of somechemical graphs as an application of generalized hierarchicalproduct of graphsrdquo Open Journal of Mathematical Sciencesvol 2 no 1 pp 338ndash350 2018

[20] N De ldquoHyper Zagreb index of bridge and chain graphsrdquoOpen Journal of Mathematical Sciences vol 2 no 1 pp 1ndash172018

[21] A T Balaban I Motoc D Bonchev and O MekenyanldquoTopological indices for structureactivity correlationsrdquo inSteric Effects in Drug Design pp 21ndash55 Springer BerlinGermany 1983

[22] I Gutman B Ruscic N Trinajstic and C F Wilcox JrldquoGraph theory and molecular orbitals XII Acyclic polyenesrdquoe Journal of Chemical Physics vol 62 no 9 pp 3399ndash34051975

[23] W Gao M Asif and W Nazeer ldquoe study of honey combderived network via topological indicesrdquo Open Journal ofMathematical Analysis vol 2 no 2 pp 10ndash26 2018

[24] H Siddiqui and M R Farahani ldquoForgotten polynomial andforgotten index of certain interconnection networksrdquo OpenJournal of Mathematical Analysis vol 1 no 1 pp 44ndash592017

[25] W Gao B Muzaffar and W Nazeer ldquoK-Banhatti and K-hyper Banhatti indices of dominating David derived net-workrdquo Open Journal of Mathematical Analysis vol 1 no 1pp 13ndash24 2017

[26] W Gao A Asghar andW Nazeer ldquoComputing degree-basedtopological indices of jahangir graphrdquo Engineering and Ap-plied Science Letters vol 1 no 1 pp 16ndash22 2018

[27] T Doslic B Furtula A Graovac I Gutman S Moradi andZ Yarahmadi ldquoOn vertex degree based molecular structuredescriptorsrdquo MATCH Communications in Mathematical andin Computer Chemistry vol 66 no 2 pp 613ndash626 2011

[28] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975

[29] M Randic ldquoOn history of the Randic index and emerginghostility toward chemical graph theoryrdquo MATCH Commu-nications in Mathematical and in Computer Chemistryvol 59 pp 5ndash124 2008

[30] M Randic ldquoe connectivity index 25 years afterrdquo Journal ofMolecular Graphics and Modelling vol 20 no 1 pp 19ndash352001

[31] K C Das and I Gutman ldquoSome properties of the secondZagreb indexrdquo MATCH Communications in Mathematicaland in Computer Chemistry vol 52 no 1 pp 3ndash1 2004

[32] L B Kier and L H HallMolecular Connectivity in Chemistryand Drug Research Academic Press New York NY USA1976

[33] L B Kier and L H HallMolecular Connectivity in Structure-Activity Analysis Research Studies Boston MA USA 1986

[34] X Li and I Gutman Mathematical Aspects of Randic-TypeMolecular Structure Descriptors University of KragujevacKragujevac Serbia 2006

[35] B Bollobas and P Erdos ldquoGraphs of extremal weightsrdquo ArsCombinatoria vol 50 pp 225ndash233 1998

[36] Y Hu X Li Y Shi T Xu and I Gutman ldquoOn moleculargraphs with smallest and greatest zeroth-order general Randicindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 54 no 2 pp 425ndash434 2005

[37] Y Ma S Cao Y Shi I Gutman M Dehmer and B FurtulaldquoFrom the connectivity index to various Randic-type de-scriptorsrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 80 no 1 pp 85ndash106 2018

[38] I Gutman B Furtula and S B Bozkurt ldquoOn Randic energyrdquoLinear Algebra and its Applications vol 442 pp 50ndash57 2014

[39] X Li and Y Shi ldquoA survey on the Randic indexrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 59 no 1 pp 127ndash156 2008

[40] M Cavers S Fallat and S Kirkland ldquoOn the normalizedLaplacian energy and general Randic index R-1 of graphsrdquoLinear Algebra and its Applications vol 433 no 1 pp 172ndash190 2010

[41] S B Bozkurt A D Gungor I Gutman and A S CevikldquoRandic matrix and Randic energyrdquo MATCH Communica-tions in Mathematical and in Computer Chemistry vol 64pp 239ndash250 2010

[42] L Pogliani ldquoFrom molecular connectivity indices to semi-empirical connectivity terms recent trends in graph theo-retical descriptorsrdquo Chemical Reviews vol 100 no 10pp 3827ndash3858 2000

[43] J B Liu M Younas M Habib M Yousaf and W NazeerldquoM-polynomials and degree-based topological indices ofVC5C7[p q] and HC5C7[p q] nanotubesrdquo IEEE Accessvol 7 pp 41125ndash41132 2019

[44] M Riaz W Gao and A Q Baig ldquoM-polynomials and degree-based topological indices of some families of convex poly-topesrdquo Open Journal of Mathematical Sciences vol 2 no 1pp 18ndash28 2018

[45] M S Anjum and M U Safdar ldquoK Banhatti and K hyper-Banhatti indices of nanotubesrdquo Engineering and AppliedScience Letters vol 2 no 1 pp 19ndash37 2019

[46] Z Shao A R Virk M S Javed M A Rehman andM R Farahani ldquoDegree based graph invariants for themolecular graph of Bismuth Tri-Iodiderdquo Engineering andApplied Science Letters vol 2 no 1 pp 01ndash11 2019

[47] L Yan M R Farahani and W Gao ldquoDistance-based indicescomputation of symmetry molecular structuresrdquo OpenJournal of Mathematical Sciences vol 2 no 1 pp 323ndash3372018

[48] R Kanabur and S K Hosamani ldquoSome numerical invariantsassociated with V-phenylenic nanotube and nanotorirdquo En-gineering and Applied Science Letters vol 1 no 1 pp 1ndash92018

[49] J B Liu C Wang and S Wang ldquoZagreb indices and mul-tiplicative Zagreb indices of eulerian graphsrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 42 pp 67ndash782019

[50] J B Liu and X F Pan ldquoAsymptotic Laplacian-energy-likeinvariant of latticesrdquo Applied Mathematics and Computationvol 253 pp 205ndash214 2015

[51] J B Liu and X-F Pan ldquoMinimizing Kirchhoffindex amonggraphs with a given vertex bipartitenessrdquo Applied Mathe-matics and Computation vol 291 pp 84ndash88 2016

[52] J B Liu J Zhao and Z Zhu ldquoOn the number of spanningtrees and normalized Laplacian of linear octagonal-quadri-lateral networksrdquo International Journal of Quantum Chem-istry vol 2019 Article ID e25971 2019

Journal of Chemistry 11

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2018

Bioinorganic Chemistry and ApplicationsHindawiwwwhindawicom Volume 2018

SpectroscopyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Medicinal ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Biochemistry Research International

Hindawiwwwhindawicom Volume 2018

Enzyme Research

Hindawiwwwhindawicom Volume 2018

Journal of

SpectroscopyAnalytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

MaterialsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

BioMed Research International Electrochemistry

International Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

[17] S Rajasekaran and S Sahni ldquoRandomized routing selectionand sorting on the OTIS-meshrdquo IEEE Transactions on Paralleland Distributed Systems vol 9 no 9 pp 833ndash840 1998

[18] A R Virk M N Jhangeer and M A Rehman ldquoReverseZagreb and reverse hyper-zagreb indices for silicon carbideSi2C3I[rs] and Si2C3II[rs]rdquo Engineering and Applied ScienceLetters vol 1 no 2 pp 37ndash50 2018

[19] N De ldquoComputing reformulated first Zagreb index of somechemical graphs as an application of generalized hierarchicalproduct of graphsrdquo Open Journal of Mathematical Sciencesvol 2 no 1 pp 338ndash350 2018

[20] N De ldquoHyper Zagreb index of bridge and chain graphsrdquoOpen Journal of Mathematical Sciences vol 2 no 1 pp 1ndash172018

[21] A T Balaban I Motoc D Bonchev and O MekenyanldquoTopological indices for structureactivity correlationsrdquo inSteric Effects in Drug Design pp 21ndash55 Springer BerlinGermany 1983

[22] I Gutman B Ruscic N Trinajstic and C F Wilcox JrldquoGraph theory and molecular orbitals XII Acyclic polyenesrdquoe Journal of Chemical Physics vol 62 no 9 pp 3399ndash34051975

[23] W Gao M Asif and W Nazeer ldquoe study of honey combderived network via topological indicesrdquo Open Journal ofMathematical Analysis vol 2 no 2 pp 10ndash26 2018

[24] H Siddiqui and M R Farahani ldquoForgotten polynomial andforgotten index of certain interconnection networksrdquo OpenJournal of Mathematical Analysis vol 1 no 1 pp 44ndash592017

[25] W Gao B Muzaffar and W Nazeer ldquoK-Banhatti and K-hyper Banhatti indices of dominating David derived net-workrdquo Open Journal of Mathematical Analysis vol 1 no 1pp 13ndash24 2017

[26] W Gao A Asghar andW Nazeer ldquoComputing degree-basedtopological indices of jahangir graphrdquo Engineering and Ap-plied Science Letters vol 1 no 1 pp 16ndash22 2018

[27] T Doslic B Furtula A Graovac I Gutman S Moradi andZ Yarahmadi ldquoOn vertex degree based molecular structuredescriptorsrdquo MATCH Communications in Mathematical andin Computer Chemistry vol 66 no 2 pp 613ndash626 2011

[28] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975

[29] M Randic ldquoOn history of the Randic index and emerginghostility toward chemical graph theoryrdquo MATCH Commu-nications in Mathematical and in Computer Chemistryvol 59 pp 5ndash124 2008

[30] M Randic ldquoe connectivity index 25 years afterrdquo Journal ofMolecular Graphics and Modelling vol 20 no 1 pp 19ndash352001

[31] K C Das and I Gutman ldquoSome properties of the secondZagreb indexrdquo MATCH Communications in Mathematicaland in Computer Chemistry vol 52 no 1 pp 3ndash1 2004

[32] L B Kier and L H HallMolecular Connectivity in Chemistryand Drug Research Academic Press New York NY USA1976

[33] L B Kier and L H HallMolecular Connectivity in Structure-Activity Analysis Research Studies Boston MA USA 1986

[34] X Li and I Gutman Mathematical Aspects of Randic-TypeMolecular Structure Descriptors University of KragujevacKragujevac Serbia 2006

[35] B Bollobas and P Erdos ldquoGraphs of extremal weightsrdquo ArsCombinatoria vol 50 pp 225ndash233 1998

[36] Y Hu X Li Y Shi T Xu and I Gutman ldquoOn moleculargraphs with smallest and greatest zeroth-order general Randicindexrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 54 no 2 pp 425ndash434 2005

[37] Y Ma S Cao Y Shi I Gutman M Dehmer and B FurtulaldquoFrom the connectivity index to various Randic-type de-scriptorsrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 80 no 1 pp 85ndash106 2018

[38] I Gutman B Furtula and S B Bozkurt ldquoOn Randic energyrdquoLinear Algebra and its Applications vol 442 pp 50ndash57 2014

[39] X Li and Y Shi ldquoA survey on the Randic indexrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 59 no 1 pp 127ndash156 2008

[40] M Cavers S Fallat and S Kirkland ldquoOn the normalizedLaplacian energy and general Randic index R-1 of graphsrdquoLinear Algebra and its Applications vol 433 no 1 pp 172ndash190 2010

[41] S B Bozkurt A D Gungor I Gutman and A S CevikldquoRandic matrix and Randic energyrdquo MATCH Communica-tions in Mathematical and in Computer Chemistry vol 64pp 239ndash250 2010

[42] L Pogliani ldquoFrom molecular connectivity indices to semi-empirical connectivity terms recent trends in graph theo-retical descriptorsrdquo Chemical Reviews vol 100 no 10pp 3827ndash3858 2000

[43] J B Liu M Younas M Habib M Yousaf and W NazeerldquoM-polynomials and degree-based topological indices ofVC5C7[p q] and HC5C7[p q] nanotubesrdquo IEEE Accessvol 7 pp 41125ndash41132 2019

[44] M Riaz W Gao and A Q Baig ldquoM-polynomials and degree-based topological indices of some families of convex poly-topesrdquo Open Journal of Mathematical Sciences vol 2 no 1pp 18ndash28 2018

[45] M S Anjum and M U Safdar ldquoK Banhatti and K hyper-Banhatti indices of nanotubesrdquo Engineering and AppliedScience Letters vol 2 no 1 pp 19ndash37 2019

[46] Z Shao A R Virk M S Javed M A Rehman andM R Farahani ldquoDegree based graph invariants for themolecular graph of Bismuth Tri-Iodiderdquo Engineering andApplied Science Letters vol 2 no 1 pp 01ndash11 2019

[47] L Yan M R Farahani and W Gao ldquoDistance-based indicescomputation of symmetry molecular structuresrdquo OpenJournal of Mathematical Sciences vol 2 no 1 pp 323ndash3372018

[48] R Kanabur and S K Hosamani ldquoSome numerical invariantsassociated with V-phenylenic nanotube and nanotorirdquo En-gineering and Applied Science Letters vol 1 no 1 pp 1ndash92018

[49] J B Liu C Wang and S Wang ldquoZagreb indices and mul-tiplicative Zagreb indices of eulerian graphsrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 42 pp 67ndash782019

[50] J B Liu and X F Pan ldquoAsymptotic Laplacian-energy-likeinvariant of latticesrdquo Applied Mathematics and Computationvol 253 pp 205ndash214 2015

[51] J B Liu and X-F Pan ldquoMinimizing Kirchhoffindex amonggraphs with a given vertex bipartitenessrdquo Applied Mathe-matics and Computation vol 291 pp 84ndash88 2016

[52] J B Liu J Zhao and Z Zhu ldquoOn the number of spanningtrees and normalized Laplacian of linear octagonal-quadri-lateral networksrdquo International Journal of Quantum Chem-istry vol 2019 Article ID e25971 2019

Journal of Chemistry 11

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Journal of

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Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom

Analytical Methods in Chemistry

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Volume 2018

Bioinorganic Chemistry and ApplicationsHindawiwwwhindawicom Volume 2018

SpectroscopyInternational Journal of

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Biochemistry Research International

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Enzyme Research

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SpectroscopyAnalytical ChemistryInternational Journal of

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MaterialsJournal of

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Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

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TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2018

Bioinorganic Chemistry and ApplicationsHindawiwwwhindawicom Volume 2018

SpectroscopyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Medicinal ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Biochemistry Research International

Hindawiwwwhindawicom Volume 2018

Enzyme Research

Hindawiwwwhindawicom Volume 2018

Journal of

SpectroscopyAnalytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

MaterialsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

BioMed Research International Electrochemistry

International Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom