cordiality of transformation graphs of cycles · 2016-07-06 · k. manjula bangalore institute of...

International Journal of Mathematics Research.ISSN 0976-5840 Volume 8, Number 2 (2016), pp. 113- 125© International Research Publication Househttp://www.irphouse.com

Cordiality of Transformation Graphs of Cycles

Chandrakala S.B.Nitte Meenakshi Institute of Technology,

Bangalore, India.

K. ManjulaBangalore Institute of Technology,

Bangalore, India.

Abstract

A graph is said to be cordial if it has a 0-1 labeling that satisfies certain properties.In this paper we show that transformation graphs of cycle are cordial. We also showthat total graph of sunlet graph, comb and star are cordial.

AMS subject classification: 05C78.Keywords: Transformation Graph, Cordial Graph.

1. Introduction

All graphs G considered here are finite, undirected and simple. We refer to [1] forunexplained terminology and notations. In 2001, Wu and Meng [3] introduced somenew graphical transformations which generalizes the concept of the total graph. As isthe case with the total graph, these generalizations referred to as transformation graphsGxyz have V (G) ∪ E(G) as the vertex set. The adjacency of two of its vertices isdetermined by adjacency and incidence nature of the corresponding elements in G.

Let α, β be two elements of V (G) ∪ E(G). Then associativity of α and β is takenas + if they are adjacent or incident in G, otherwise −. Let xyz be a 3-permutation ofthe set {+, −}. The pair α and β is said to correspond to x or y or z of xyz if α andβ are both in V (G) or both are in E(G), or one is in V (G) and the other is in E(G)

respectively. Thus the transformation graph Gxyz of G is the graph whose vertex set isV (G) ∪ E(G). Two of its vertices α and β are adjacent if and only if their associativityin G is consistent with the corresponding element of xyz.

In particular the transformation graph G++− of G is the graph with vertex set V (G)∪E(G) in which the vertices u and v are joined by an edge if one of the following holds

114 Chandrakala S.B. and K. Manjula

1. both u, v ∈ V (G) and u and v are adjacent in G

2. both u, v ∈ E(G) and u and v are adjacent in G

3. one is in V (G) and the other is in E(G) and they are not incident with each otherin G.

The transformation graphs are investigated in [4], [5] and [6].A graph labeling is an assignment of integers to the vertices or edges, or both,

subject to certain conditions. A mapping f : V (G) → {0, 1} is called a binary labelingof the graph G. For each v ∈ V (G), f (v) is called the vertex label of the vertex v

under f and for an edge uv the induced edge labeling g : E(G) → {0, 1} is given byg(uv) = |f (u) − f (v)|. Then f is called a cordial labeling of G if the number ofvertices labeled 0 and the number of vertices labeled 1 differs by at most 1, and, thenumber of edges labeled 0 and the number of edges labeled 1 differs by at most 1. Agraph G is cordial if it admits a cordial labeling. This concept is introduced by Cahit[2]. For more cordial graphs we refer to [8], [9], [10], [12] and [11].

For convenience, the transformation graph Gxyz is partitioned into Gxyz = Sx(G) ∪Sy(G)∪Sz(G) where Sx(G), Sy(G) and Sz(G) are the edge-induced subgraphs of Gxyz.The edge set of each of which is respectively determined by x, y and z of the permutationxyz. Sx(G) ∼= G when x is + and Sx(G) ∼= G when x is −. Sy(G) ∼= L(G) when y is+ and Sy(G) ∼= L(G) when y is −. When z is +, α, β ∈ V (Gxyz) are adjacent in Sz(G)

if they are incident with each other in G. When z is −, α, β are adjacent in Sz(G) if theyare not incident in G.

The following notations are used in relation to labeling of Gxyz:Let V0 and V1 denote the set of vertices of Gxyz labeled 0 and 1.E0 and E1 denote set of edges of Gxyz labeled 0 and 1 respectively.E0(Sx) and E1(Sx) denote set of edges labeled 0 and 1 in Sx(G). Similar meanings

are associates with E0(Sy), E1(Sy), E0(Sz) and E1(Sz) .For P xyz

n xyz ∈ {+ + −, + − −, − + +, − + −, − − +, − − −, + + +} wedefine a vertex labeling f : V (P xyz

n ) → {0, 1} and specify the induced edge labelingg : E(P xyz

n ) → {0, 1} then show that f is a cordial labeling.

2. Cordiality of Transformation Graphs of Cycle

Let Cn : v1 − v2 − v3 − . . . − vn(n ≥ 3) be the cycle on n vertices and ei = vivi+1(1 ≤i ≤ n − 1) and en = vnv1 be the edges of Cn.

Theorem 2.1. For any positive integer n ≥ 3

(a) C++−n is cordial

(b) C+++n is cordial

(c) C−++n is cordial when n ≡ 0, 2, 3, 5, 6, 7(mod 8)

Cordiality of Transformation Graphs of Cycles 115

(d) C+−−n is cordial when n ≡ 0, 1, 2, 3, 5, 6(mod 8)

(e) C+−+n is cordial when n ≡ 0, 2, 3, 5, 6, 7(mod 8)

(f) C−−−n is Cordial when n ≡ 0, 1, 3, 4, 5, 7(mod 8)

(g) C−+−n is Cordial when n ≡ 0, 1, 2, 3, 5, 6(mod 8)

(h) C−−+n is Cordial when n ≡ 0, 1(mod 4).

Proof.

(a) For n ≥ 3 the vertices of C−+−n are labeled as in table 1:

n )( ivf )( ief 10 VV −

Even

−≤≤

otherwise

ni

,0

12

1,1

+≤≤

otherwise

ni

,0

12

1,1

0

Odd

≤≤

otherwise

ni

,02

1,1

≤≤

otherwise

ni

,02

1,1

Table 1: Cordial labeling of C++−n .

|E0| and |E1| are calculated from the labeling of the vertices of V (C++−n ), and are

given in the table 2:

n )(0 xSE )(0 ySE )(0 zSE � )(1 xSE )(1 ySE )(1 zSE 10 ~ EE

Even

22

−n

22

−n

4

)2(2

−+−

nnn

4

)4)(2( −++

nn

2 2

4

)2(2

−+−

nnn

4

)4)(2( −++

nn 0

Odd 2

742+− nn

2

72−n

1

Table 2: |E0| and |E1| of C++−n .

Therefore C++−n is cordial.

(b) The vertices of C+++n are labeled as below:

f (vi) = 1 for 1 ≤ i ≤ n

f (ei) = 1 for 1 ≤ i ≤ n


Here |V0| − |V1| = 0|E0| and |E1| are calculated from the labeling of the vertices of C+++

n , and are:

|E0| = |E0(Sx)| + |E0(Sy)| + |E0(Sz)| = n + n + 0 = 2n

|E1| = |E1(Sx)| + |E1(Sy)| + |E1(Sz)| = 0 + 0 + 2n = 2n

Therefore |E0| − |E1| = 0.

Therefore C+++n is cordial.

(c) When 3 ≤ n ≤ 7, the vertices of C−++n are labeled as in Table 3 which admits

cordial labeling.

n f (v1)f (v2)...f (vn) f (e1)f (e2)...f (en)

3 101 1004 1010 11005 10101 110006 101010 1100107 1010101 1100100

Table 3: Cordial labeling of C−++n for the case n < 7.

when n ≥ 8, the vertex labeling of C−++n are given in Table 4 and |E0| and |E1|

of C−++n are counted in Table 5.

n )( ivf )( ief 10 ~ VV

rn 8=

28 += rn 1

≤≡≤+

≤≡≤

otherwise

nir

ri

1)2(mod0240

4)4(mod3,030

0 38 += rn

58 += rn78 += rn

�

0

=

−≤≡≤+

+≤≡≤

otherwise

ni

nir

ri

10

1)2(mod0640

44)4(mod3,030

�

68 += rn1

≤≡≤+

+≤≡≤

otherwise

nir

ri

1)2(mod0640

44)4(mod3,030

Table 4: cordial labeling of C−++n for the case n ≥ 8.

Therefore C−++n is cordial.

(d) For n = 3, 5and6, the vertices of C+−−n are labeled as in Table 6 which admits

cordial labeling.

For n ≥ 8, the vertices of C+−−n are labeled as in Table 7. In all the cases

|V0| − |V1| = 0.


n )(0 xSE )(0 ySE )(0 zSE )(1 xSE )(1 ySE )(1 zSE 10 ~ EE

�

rn 8=

∑

−

=

2

2

1

2

n

i

i r2 n2

42

2

4

1

−+∑

−

=

ni

n

i

)2( rn − n

0

28 += rn1

38 += rn

2

32

2

3

1

−+∑

−

=

ni

n

i

32 +r 1−n ∑

−

=

2

3

1

2

n

i

i32 −− rn

1+n

1

58 += rn0

78 += rn1

68 += rn

∑

−

=

2

2

1

2

n

i

i 22 +r n2

42

2

4

1

−+∑

−

=

ni

n

i

22 −− rnn 1

Table 5: |E0| and |E1| of C−++n .

n f (v1)f (v2)...f (vn) f (e1)f (e2)...f (en)

3 110 0015 11000 101016 110010 101010

Table 6: Cordial labeling of C+−−n for the case n < 8.

|E0| and |E1| of C+−−n for the case n ≥ 8 are given Table 8.

From the Table 8 it is evident that C+−−n .

(e) For 3 ≤ n ≤ 8, the vertices of C+−+n are labeled as in Table 9 which admits cordial

labeling.

For n ≥ 8, f is defined as in (d) which admits cordial labeling.

(f) For n = 3, 4, 5, 7, the vertices of C−−−n are labeled as in Table 10 which admits

cordial labeling.

For n ≥ 8, the vertices of C−−−n are labeled as in Table 11 and |E0| and |E1| of

C−−−n for the case n ≥ 8 are given in Table 12.

Therefore C−−−n is cordial.

(g) For 3 ≤ n ≤ 8, the vertices of C−+−n are labeled as in Table 13 which admits cordial

labeling.

For n ≥ 8, the vertices of C−+−n labeled as in Table 14 and |E0| and |E1| of C−+−

n

for the case n ≥ 8 are given table 15:

Therefore C−+−n is cordial.


n )( ief )( ivf10 ~ VV

rn 8=

28 += rn 0

≤≡≤+

≤≡≤

otherwisenir

ri

1)4(mod2,0240

4)4(mod3,030�

0 ,18 += rn,38 += rn

58 += rn��

1

=

−≤≡≤+

+≤≡≤

otherwiseni

nir

ri

10

1)4(mod2,0640

44)4(mod3,030

�68 += rn0

Table 7: Cordial labeling of C+−−n for the case n ≥ 8.

n )(0 xSE

)(0 ySE )(0 zSE )(1 xSE )(1 ySE )(1 zSE 10 ~ EE

�

rn 8=

r2 ∑

−

=

2

2

1

2

n

i

i

( ) +− rn 2

)2(22

rnn

−

− rn 2− 2

42

2

4

1

−+∑

−

=

ni

n

i

( ) +− rn 2

)2(22

rnn

−

−

0

28 += rn1

38 += rn

32 +r 2

32

2

3

1

−+∑

−

=

ni

n

i 2

322−− nn 32 −− rn

∑

−

=

2

3

1

2

n

i

i 2

322+− nn

1

58 += rn0

78 += rn1

68 += rn22 +r ∑

−

=

2

2

1

2

n

i

i 2

)2( −nn 22 −− rn2

42

2

4

1

−+∑

−

=

ni

n

i 2

)2( −nn1

Table 8: |E0| and |E1| of C+−−n for the case n ≥ 8

(h) Vertices of V (C−−+n ) are as in Table 16 and |E0| and |E1| of C−−+

n for the casen ≥ 8 are given table 17:

�

3. Cordiality of G+++ when G isomorphic to Sunlet,Comb and Star Graphs

We recall the following definitions:

Definition 3.1. Corona G1 �G2 of G1 and G2 is the graph obtained by taking one copyof G1(which has n1 vertices) and n1 copies of G2 and then joining ith vertex of G1 toevery vertex in the ith copy of G2. So the order G1 � G2 is n1 + n1n2 where n2 is theorder of G2 and its size is m1 + n1n2 + n1m2 where mi is the size of Gi , ∀ i = 1, 2.


n f (v1)f (v2) · · · f (vn) f (e1)f (e2) · · · f (en)

3 001 0114 1101 10005 11001 010106 110010 0101017 1100101 0101010

Table 9: Cordial labeling ofC+−+n for the case 3 ≤ n ≤ 8.


3 010 1104 0101 11005 01010 110017 0101010 1100101

Table 10: Cordial labeling ofC−−−n for the case n < 8.

Definition 3.2. Pn � K1 (n ≥ 3) is called comb graph of order 2n.Definition 3.3. Cn � K1 (n ≥ 3) is called sunlet graph of order 2n.

Theorem 3.4. For any positive integer n ≥ 3 total graph of sunlet graph Sn is cordial.

Proof. Let vi(1 ≤ i ≤ n) be the consecutive vertices on the cycle Cn, v′i be the pendant

vertices adjacent to vi respectively. Let ei = vivi+1 be the edges on the cycle and e′i be

the pendant edges incident to vi respectively.Define a binary labeling f : V (S+++

n ) → {0, 1} as in Table 18 and |E0| and |E1| ofS+++

n for the case n ≥ 6 are given Table 19:Therefore S+++

n is cordial. �

n )( ivf )( ief 10 ~ VV

)8(mod4,0≡n �

evenisi

oddisi

1

0

≡

otherwise

i

0

)4(mod2,11

0

)8(mod5,1≡n

)8(mod7≡n

−=

−=

−≤≡

−≤≡

10

,21

3)4(mod3,00

5)4(mod2,11

ni

nni

ni

ni

�

)8(mod3≡n

=

>

evenisii

oddisandi

111

10�

≡

otherwise

i

0

)4(mod4,31

Table 11: cordial labeling of C−−−n for the case n ≥ 8.



�

)8(mod

4,0≡n

∑

−

=

2

2

1

2

n

i

i ∑

−

=

4

4

1

8

n

i

i 2

)2( −nn2

42

2

4

1

−+∑

−

=

ni

n

i ∑

−

=

2

2

1

2

n

i

i 2

)2( −nn0

)8(mod

5,1≡n

2

32

2

3

1

−+∑

−

=

ni

n

i

122

3

1

−∑

−

=

n

i

i

2

)1( 2−n

∑

−

=

2

3

1

2

n

i

i

2

12

2

3

1

−+∑

−

=

ni

n

i

2

122−− nn

1

)8(mod

7≡n

∑

−

=

2

3

1

2

n

i

i 2

32

2

3

1

−+∑

−

=

ni

n

i

1

)8(mod

3≡n 1

Table 12: |E0| and |E1| of C−−−n for the case n ≥ 8


3 101 1004 11100 00015 10101 000116 10101 1101017 1010101 1101000

Table 13: Cordial labeling ofC−+−n for the case 3 ≤ n ≤ 8.

Theorem 3.5. Total graph of Comb graph is cordial.

Proof. Let G = Pn

⊙K1(n ≥ 3) be a comb on 2n vertices. Let vi(1 ≤ i ≤ n) be the

consecutive vertices of degree 2, v′i(1 ≤ i ≤ n) be the pendant vertices adjacent to vi

respectively. Let ei = vivi+1(1 ≤ i ≤ n − 1) be the edges and e′i(1 ≤ i ≤ n) be the

pendant edges incident to vi respectively. Define a binary labeling f : V (G) → {0, 1}as in Table 20 and |E0| and |E1| of total graph of comb for the case n ≥ 6 are as shownin Table 21.

�

n )( ivf )( ief 10 ~ VV

n=8r,

8r+2

evenisi

oddisi

0

1

≤≡≤+

≤≡

otherwise

nir

ri

1)4(mod4,2240

4)4(mod0,30

0 n=8r+6

n=8r+1

8r+3,

8r+5

≤≡≤+

+≤≡

otherwise

nir

ri

1)4(mod4,2640

44)4(mod0,30

Table 14: cordial labeling of C−+−n for the case n ≥ 8.



)8(mod0≡n

∑

−

=

2

2

1

2

n

i

i r22

)2( −nn

2

42

2

4

1

−+∑

−

=

ni

n

i

rn 2− 2

)2( −nn 0

)8(mod2≡n1

)8(mod1≡n

2

32

2

3

1

−+∑

−

=

ni

n

i

12 +r2

)1( 2−n

∑

−

=

2

3

1

2

n

i

irn 21−−

2

122−− nn

1

)8(mod3≡n0

)8(mod5≡n1

)8(mod6≡n

∑

−

=

2

2

1

2

n

i

i 22 +r2

)2( −nn

2

42

2

4

1

−+∑

−

=

ni

n

i

rn 22 −−

2

)2( −nn

1

Table 15: |E0| and |E1| of C−+−n for the case n ≥ 8

n )( ivf )( ief 10 VV −

)4(mod0≡n

≡

otherwise

i

1

)4(mod3,00

0 )4(mod1≡n

≡

otherwise

i

1

)4(mod3,00

=

−≤≡≤

otherwise

ni

ni

1

,3,20

2)4(mod3,060

Table 16: Cordial labeling of C−−+n .

Theorem 3.6. Total graph of Star K1,n wheren ≡ 0, 1, 3, 4, 5, 6(mod 8) is cordial.

Proof. Let v0 be the central vertex and vi(1 ≤ i ≤ n) be the pendant vertices of K1,n.

Let ei = v0vi(1 ≤ i ≤ n) be the pendant edges.Define a binary labeling f : V (G) → {0, 1} as in Table 22 and |E0| and |E1| of total

graph of Star for the case n ≥ 6 are as shown in Table 23. �

4. Conclusion

We did not get the cordial labeling of Gxyz for some specific values of n when G isisomorphic to path, cycle and star. One of the future work on this paper is to find somemore standard graphs G for which the transformation graphs are cordial.


n )(0 xSE )(0 ySE )(0 zSE )(1 xSE )(1 ySE )(1 zSE 10 EE −

�

)4(mod0≡n

∑

−

=

4

4

1

8

n

i

i ∑

−

=

4

4

1

8

n

i

i 2

nn + ∑

−

=

2

2

1

2

n

i

i ∑

−

=

2

2

1

2

n

i

i 2

n0

)4(mod1≡n ∑

−

=

+−

2

5

1

2)4(

n

i

in

+− rn 2

∑=

−

r

i

i1

)12(42

1−n ∑

−

=

+−

2

3

1

22

1n

i

in

122

3 2

3

1

−+−

∑

−

=

n

i

in 2

2

1+

−n

0

Table 17: |E0| and |E1| of C−−+n

)6( ≥n )( ivf )( ivf ′ )( ief )( ief ′10 VV −

rn 4=

0

)1( ni ≤≤

1

)1( ni ≤≤

≤≤

otherwise

ri

,0

1,1

≤≤

otherwise

ri

,1

1,00 14 += rn

34 += rn +≤≤

otherwise

ri

,0

11,1

+≤≤

otherwise

ri

,1

11,0

Table 18: Cordial labeling of S+++n for the case n ≥ 6.

)6( ≥n )( 10 SE )( 20 SE )( 30 SE )( 11 SE )( 21 SE )( 31 SE 10 EE − �

n=4r, 4r+1 n n 3n-2r n 2n n+2r 0, 1

34 += rn n n 3n-2r-2 n 2n n+2r+2 -1

Table 19: |E0| and |E1| of S+++n

for the case n ≥ 6


)6( ≥n )( ivf )( ivf ′ )( ief )( ief ′ 10 VV −

rn 4=

0

)1( ni ≤≤

1

)1( ni ≤≤

−≤≤−

otherwise

nir

1

210

−≤≤−

otherwise

nir

0

111

0 14 += rn

−=

≤≤

otherwiseni

ri

011

21

=

≤≤

otherwiseni

ri

10

10

24 += rn

−=

−≤≤

otherwiseni

ri

011

111

=

−≤≤

otherwiseni

ri

10

110

34 += rn

Table 20: Cordial labeling of total graph of Comb for the case n ≥ 6.

)6( ≥n )(0 xSE )(0 ySE)(0 zSE

)(1 xSE )(1 ySE )(1 zSE 10 ~ EE

�

rn 4=

n-1

n-2 3n-2r

n

2n-2 n+2r-2 1

14 += rn n-1

3n-2r-2

2n-3

n+2r

0

24 += rnn-2 2n-2

1

34 += rn 0

Table 21: |E0| and |E1| of total graph of comb for the case n ≥ 6.

)6( ≥n )( ivf )( ief 10 ~ VV

rn 8=

≤≤

otherwise

ni

,0

21,1

≤≤++

−

≤≤

otherwise

nirn

andrn

i

,0

12

21,1

1

18 += rn1

28 += rn1

38 += rn1

48 += rn1

58 += rn1

68 += rn

≤≤++

−−

≤≤

otherwise

nirn

andrn

i

,0

22

12

1,1

1

Table 22: Cordial labeling of total graph of Star for the case n ≥ 6.


)6( ≥n )(0 xSE )(0 ySE )(0 zSE )(1 xSE )(1 ySE )( 31 SE 10 ~ EE

�

rn 8=

2

n

×

2

2

2

2

nnrn

n22

2−+

2

n

×

22

nn

rn

22

+

0

18 += rn 1

28 += rn 1

38 += rn 0

48 += rn 0

58 += rn 1

68 += rn 222

3−− r

n 222

++ rn 1

Table 23: |E0| and |E1| of total graph of Star for the case n ≥ 6.

Acknowledgements

The authors wish to express their gratitude to Dr. B. Sooryanarayana for his helpfulcomments and suggestions.

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cordiality of transformation graphs of cycles · 2016-07-06 · k. manjula bangalore institute of...

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