new results on the energy of integral circulant graphs

Applied Mathematics and Computation 218 (2011) 3470–3482

Contents lists available at SciVerse ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

New results on the energy of integral circulant graphs

Aleksandar Ilic, Milan Bašic ⇑Faculty of Sciences and Mathematics, Višegradska 33, 18000 Niš, Serbia

a r t i c l e i n f o a b s t r a c t

Keywords:Integral circulant graphsGraph energyEigenvaluesCospectral graphs

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.08.094

⇑ Corresponding author.E-mail addresses: [email protected] (A. Ilic

Circulant graphs are an important class of interconnection networks in parallel and distrib-uted computing. Integral circulant graphs play an important role in modeling quantumspin networks supporting the perfect state transfer as well. The integral circulant graphICGn(D) has the vertex set Zn = {0,1,2, . . . ,n � 1} and vertices a and b are adjacent ifgcd(a � b,n) 2 D, where D # {d : djn,1 6 d < n}. These graphs are highly symmetric, haveintegral spectra and some remarkable properties connecting chemical graph theory andnumber theory. The energy of a graph was first defined by Gutman, as the sum of the abso-lute values of the eigenvalues of the adjacency matrix. Recently, there was a vast researchfor the pairs and families of non-cospectral graphs having equal energies. Following Bapatand Pati [R.B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bull. Kerala Math.Assoc. 1 (2004) 129–132], we characterize the energy of integral circulant graph modulo4. Furthermore, we establish some general closed form expressions for the energy of inte-gral circulant graphs and generalize some results from Ilic [A. Ilic, The energy of unitaryCayley graphs, Linear Algebra Appl. 431 (2009), 1881–1889]. We close the paper by pro-posing some open problems and characterizing extremal graphs with minimal energyamong integral circulant graphs with n vertices, provided n is even.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

Circulant graphs are Cayley graphs over a cyclic group. The interest of circulant graphs in graph theory and applicationshas grown during the last two decades, they appeared in coding theory, VLSI design, Ramsey theory and other areas. Recentlythere is vast research on the interconnection schemes based on circulant topology–circulant graphs represent an importantclass of interconnection networks in parallel and distributed computing (see [21]). Integral circulant graphs are also highlysymmetric and have some remarkable properties between connecting graph theory and number theory.

In quantum communication scenario, circulant graphs is used in the problem of arranging N interacting qubits in a quan-tum spin network based on a circulant topology to obtain good communication between them. In general, quantum spin sys-tem can be defined as a collection of qubits on a graph, whose dynamics is governed by a suitable Hamiltonian, withoutexternal control on the system. Different classes of graphs were examined for the purpose of perfect transferring the statesof the systems. Since circulant graphs are mirror symmetric, they represent good candidates for the property of periodicityand thus integrality [12], which further implies that integral circulant graphs would be potential candidates for modeling thequantum spin networks that permit perfect state transfer [1–3,14,33]. These properties are primarily related to the spectra ofthese graphs. Indeed, the eigenvalues of the graphs are indexed in palindromic order (ki = kn�i) and can be represented byRamanujan’s sums.

. All rights reserved.

), [email protected] (M. Bašic).

http://dx.doi.org/10.1016/j.amc.2011.08.094

mailto:[email protected]

mailto:[email protected]

http://dx.doi.org/10.1016/j.amc.2011.08.094

http://www.sciencedirect.com/science/journal/00963003

http://www.elsevier.com/locate/amc

A. Ilic, M. Bašic / Applied Mathematics and Computation 218 (2011) 3470–3482 3471

Bašic [7,8] established a condition under which integral circulant graphs have perfect state transfer and gave completecharacterization these graphs. It turned out that the degree of 2 must be equal in a prime factorization of the differenceof successive eigenvalues. Furthermore, exactly one of the divisors n/4 or n/2 have to belong to the divisor set D for any inte-gral circulant graph ICGn(D) having perfect state transfer. In this paper we continue with studying parameters of integralcirculant graphs like energy, having in mind application in chemical graph theory. We actually focus on characterizationof the energy of integral circulant graphs ICGn(D) modulo 4, where the divisor n/2 and eigenvalue kn/2 play important role.During this task, some interesting properties of the eigenvalues modulo 2 are also used.

Saxena et al. [33] studied some parameters of integral circulant graphs as the bounds for the number of vertices and thediameter, bipartiteness and perfect state transfer. The present authors in [6,23] calculated the clique and chromatic numberof integral circulant graphs with exactly one and two divisors, and also disproved posed conjecture that the order of ICGn (D)is divisible by the clique number. Klotz and Sander [26] determined the diameter, clique number, chromatic number andeigenvalues of the unitary Cayley graphs. The latter group of authors proposed a generalization of unitary Cayley graphsnamed gcd-graphs and proved that they have to be integral.

Let A be the adjacency matrix of a simple graph G, and k1,k2, . . . ,kn be the eigenvalues of the graph G. The energy of G isdefined as the sum of absolute values of its eigenvalues [15,16,19]

EðGÞ ¼Xn

i¼1

jkij:

The concept of graph energy arose in chemistry where certain numerical quantities, such as the heat of formation of ahydrocarbon, are related to total p-electron energy that can be calculated as the energy of an appropriate molecular graph.

The graph G is said to be hyperenergetic if its energy exceeds the energy of the complete graph Kn, or equivalently ifE(G) > 2n � 2. This concept was introduced first by Gutman and afterwards has been studied intensively in the literature[4,9,17,35]. Hyperenergetic graphs are important because molecular graphs with maximum energy pertain to maximalitystable p-electron systems. In [22,31], the authors calculated the energy of unitary Cayley graphs and complement of unitaryCayley graphs, and establish the necessary and sufficient conditions for ICGn to be hyperenergetic. There was a vast researchfor the pairs and families of non-cospectral graphs having equal energy [10,11,24,25,27,28,30,36].

In 2004 Bapat and Pati [5] proved an interesting simple result–the energy of a graph cannot be an odd integer. Pirzada andGutman [29] generalized this result and proved the following

Theorem 1.1. Let r and s be integers such that r P 1 and 0 6 s 6 r � 1. Let q be an odd integer. Then E(G) cannot be of the form(2sq)1/r.

For more information about the closed forms of the graph energy we refer the reader to [32].In this paper we go to a step further and characterize the energy of integral circulant graph modulo 4.The paper is organized as follows. In Section 2 we give some preliminary results regarding eigenvalues of integral circ-

ulant graphs. In Section 3 we characterize the energy of integral circulant graph modulo 4, while in Section 4 we generalizedformulas for the energy of integral circulant graphs from [22]. In Section 5, some larger families of graphs with equal energyare presented and further we support conjecture proposed by So [34], that two graphs ICGn(D1) and ICGn(D2) are cospectral ifand only if D1 = D2. In concluding remarks we propose some open problems and characterize extremal graphs with minimalenergy among integral circulant graphs with n vertices, provided n is even.

2. Preliminaries

Let us recall that for a positive integer n and subset S # {0,1,2, . . . ,n � 1}, the circulant graph G(n,S) is the graph with nvertices, labeled with integers modulo n, such that each vertex i is adjacent to jSj other vertices {i + s (mod n)js 2 S}. The set Sis called a symbol of G(n,S). As we will consider only undirected graphs without loops, we assume that 0 R S and, s 2 S if andonly if n � s 2 S, and therefore the vertex i is adjacent to vertices i ± s (mod n) for each s 2 S.

Recently, So [34] has characterized circulant graphs with integral eigenvalues–integral circulant graphs. Let

GnðdÞ ¼ fkj gcdðk;nÞ ¼ d;1 6 k < ng;

be the set of all positive integers less than n having the same greatest common divisor d with n. Let Dn be the set of positivedivisors d of n, with d 6 n

2.

Theorem 2.1. A circulant graph G(n,S) is integral if and only if

S ¼[d2D

GnðdÞ;

for some set of divisors D # Dn.We denote them by ICGn(D) and in some recent papers integral circulant graphs are also known as gcd-graphs [6,26].Let C be a multiplicative group with identity e. For S � C, e R S and S�1 = {s�1js 2 S} = S, the Cayley graph X = Cay(C,S) is the

undirected graph having vertex set V(X) = C and edge set E(X) = {{a,b}jab�1 2 S}. For a positive integer n > 1 the unitary

3472 A. Ilic, M. Bašic / Applied Mathematics and Computation 218 (2011) 3470–3482

Cayley graph Xn = Cay(Zn,Un) is defined by the additive group of the ring Zn of integers modulo n and the multiplicative groupUn ¼ Z�n of its invertible elements.

By Theorem 2.1 we obtain that integral circulant graphs are Cayley graphs of the additive group of Zn with respect to theCayley set S =

Sd2DGn(d). From Corollary 4.2 in [21], the graph ICGn(D) is connected if and only if gcd(d1,d2, . . . ,dk) = 1.

Let A be a circulant matrix. The entries a0,a1, . . . ,an� 1 of the first row of the circulant matrix A generate the entries of theother rows by a cyclic shift (for more details see [13]). There is an explicit formula for the eigenvalues kk, 0 6 k 6 n � 1, of acirculant matrix A. Define the polynomial Pn(z) by the entries of the first row of A,

PnðzÞ ¼Xn�1

i¼0

ai � zi:

The eigenvalues of A are given by

kj ¼ PnðxjÞ ¼Xn�1

i¼0

ai �xji; 0 6 j 6 n� 1; ð1Þ

where x = exp(ı2p/n) is the nth root of unity. Ramanujan’s sum [38], usually denoted c(k,n), is a function of two positiveinteger variables n and k defined by the formula

cðk;nÞ ¼Xn

a¼1gcdða;nÞ¼1

e2pin �ak ¼

Xn

a¼1gcdða;nÞ¼1

xakn ;

where xn denotes a complex primitive nth root of unity. These sums take only integral values,

cðk;nÞ ¼ lðtn;kÞ �uðnÞuðtn;kÞ

where tn;k ¼n

gcdðk;nÞ

and l denotes the Möbious function. In [26] it was proven that gcd-graphs (the same term as integral circulant graphsICGn(D)) have integral spectrum,

kk ¼Xd2D

c k;nd

� �; 0 6 k 6 n� 1: ð2Þ

Using the well-known summation [20]

sðk;nÞ ¼Xn�1

i¼0

xikn ¼

0 if n-kn if njk

8><>: ;

we get that

Xn�1

k¼0

cðk;nÞ ¼ 0: ð3Þ

For even n it follows

Xn=2�1

k¼0

cðk;nÞ ¼Xn

a¼1gcdða;nÞ¼1

Xn=2�1

k¼0

xakn ¼

Xn=2

a¼1gcdða;nÞ¼1

Xn=2�1

k¼0

xakn þxðn�aÞk

n

!

¼Xn=2

a¼1gcdða;nÞ¼1

xann �xan=2

n þXn�1

k¼0

xakn

!¼

Xn=2

a¼1gcdða;nÞ¼1

ð1þ 1Þ ¼ uðnÞ: ð4Þ

Similarly, for odd n it follows

Xðn�1Þ=2

k¼0

cðk;nÞ ¼ uðnÞ2

: ð5Þ

It also follows that if k � k0 (mod n) then c(k,n) = c(k0,n).Throughout the paper, we let n ¼ pa1

1 pa22 � � � p

akk , where p1 < p2 < � � � < pk are distinct primes, and ai P 1.

3. The energy of integral circulant graphs modulo 4

Note that for arbitrary divisor d and 1 6 i 6 n � 1, it holds


tn=d;i ¼n=d

gcdðn=d; iÞ ¼n

gcdðn; idÞ

and

tn=d;n�i ¼n=d

gcdðn=d;n� iÞ ¼n

gcdðn;nd� idÞ :

Since gcd(n, id) = gcd(n,nd � id), we have tn/d,i = tn/d,n�i. Finally,

cði;n=dÞ ¼ lðtn=d;iÞuðn=dÞuðtn=d;iÞ

¼ lðtn=d;n�iÞuðn=dÞ

uðtn=d;n�iÞ¼ cðn� i;n=dÞ;

for each 1 6 i 6 n � 1. Therefore we have the following assertion.

Lemma 3.1. Let ICGn(D) be an arbitrary integral circulant graph. Then for each 1 6 i 6 n � 1, the eigenvalues ki and kn�i of ICGn(D)are equal.

For i = 0 we have

k0 ¼Xd2D

uðn=dÞ;

while for n even and i = n/2 we have

kn=2 ¼Xd2D

ð�1Þduðn=dÞ:

3.1. Energy modulo 4 for n odd

According to Lemma 3.1, the energy of G ffi ICGn(D) is equal to

EðGÞ ¼ k0 þ 2Xðn�1Þ=2

i¼1

jkij:

Since x � jxj (mod 2), in order to characterize E(G) modulo 4 we consider the parity of the following sum

EðGÞ2�X

d2D

uðn=dÞ2

þXðn�1Þ=2

i¼1

Xd2D

cði;n=dÞðmod 2Þ:

Since n/d > 2, it follows that u(n/d) is even. After exchanging the order of the summation we have

EðGÞ2�X

d2D

uðn=dÞ2

þXd2D

Xðn�1Þ=2

i¼1

cði;n=dÞðmod 2Þ: ð6Þ

By relation (3), we get that for every k it holds that

Xkþn�1

i¼k

cði;nÞ ¼ 0: ð7Þ

Theorem 3.2. For odd n, the energy of ICGn(D) is divisible by four.

Proof. Using the following relation n�12 ¼ n

d � d�12 þ n�d

2d , the formula for graph energy (6) now becomes

EðGÞ2�X

d2D

uðn=dÞ2

þXd2D

Xd�12 �1

l¼0

Xðlþ1Þnd

i¼l ndþ1

cði; n=dÞ þXn�1

2

i¼nðd�1Þ2d þ1

cði;n=dÞ

0B@

1CAðmod 2Þ: ð8Þ

Next we get

EðGÞ2�X

d2D

uðn=dÞ2

þXðn=d�1Þ=2

i¼1

cði; n=dÞðmod 2Þ

and using relation (5), we get that


EðGÞ2�X

d2D

uðn=dÞ2

þuðn=dÞ2

�uðn=dÞ � 0ðmod 2Þ:

This implies that 4jE(G). h

3.2. Energy modulo 4 for n even

According to Lemma 3.1, the energy of G ffi ICGn(D) is equal to

EðGÞ ¼ jk0j þ jkn=2j þ 2Xn=2�1

i¼1

jkij:

Using the same reasoning as in the previous subsection, we get that k0 and kn/2 are of the same parity,

jk0j þ jkn=2j ¼Xd2D

uðn=dÞ þXd2D

ð�1Þduðn=dÞ��

��:
Also,
S ¼ 12� ðjk0j þ jkn=2jÞ ¼

Xd2D

uðn=dÞ2

þX

d2Dð�1Þd uðn=dÞ

2

�� ¼

Pd2D; d even

uðn=dÞ; if kn=2 > 0

Pd2D; d odd

uðn=dÞ; if kn=2 < 0

8><>: :

If n2 R D, then 2ju(n/d) and S � 0 (mod 2); otherwise we conclude that

S �0; if kn=2 > 0 and 4-n; or kn=2 < 0 and 4jn ðmod 2Þ1; if kn=2 > 0 and 4jn; or kn=2 < 0 and 4-n ðmod 2Þ

�:

Therefore

EðGÞ2� Sþ

Xd2D

Xn=2�1

i¼1

cði;n=dÞ ðmod 2Þ:

Theorem 3.3. For even n, the energy of ICGn(D) is not divisible by four if and only if n2 R D and kn/2 is negative.

Proof. If d is even, we have n2� 1 ¼ d

2 � nd� 1. Since c(0,n/d) = u(n/d), it follows

Xn=2�1

i¼1

cði;n=dÞ ¼ �cð0; n=dÞ þXd=2

k¼1

Xk�n=d�1

i¼ðk�1Þ�n=d

cði;n=dÞ ¼ �uðn=dÞ þ d2�Xn=d�1

i¼0

cði;n=dÞ ¼ �uðn=dÞ:

If d is odd, we have n2� 1 ¼ d�1

2 � ndþ 1

2 � nd� 1. Similarly, using the relation (4), it follows

Xn=2�1

i¼1

cði;n=dÞ ¼ �cð0; n=dÞ þXðd�1Þ=2

k¼1

Xk�n=d�1

i¼ðk�1Þ�n=d

cði;n=dÞ þXn=2�1

i¼ððd�1Þ=2Þ�n=d

cði;n=dÞ

¼ �uðn=dÞ þ d� 12�Xn=d�1

i¼0

cði;n=dÞ þXn=ð2dÞ�1

i¼0

cði;n=dÞ ¼ �uðn=dÞ þuðn=dÞ ¼ 0:

For n2 R D, we have that S � 0 (mod 2) and 4jE(G).

For n2 2 D, by combining above cases we have

Xd2D

Xn=2�1

i¼1

cði;n=dÞ � 1þ ð�1Þn=2

2ðmod 2Þ:

For kn/2 > 0, it follows

EðGÞ2� Sþ 1þ ð�1Þn=2

2� 1þ ð�1Þn=2

2þ 1þ ð�1Þn=2

2� 0 ðmod 2Þ;

while for kn/2 < 0, we have

EðGÞ2� Sþ 1þ ð�1Þn=2

2� 1� ð�1Þn=2

2þ 1þ ð�1Þn=2

2� 1 ðmod 2Þ:

This completes the proof. h


4. The energy of some classes of integral circulant graphs

Here we generalize results from [22].

Theorem 4.1. Let n P 4 be an arbitrary integer. Then the energy of the integral circulant graph Xn(1,pc) for c P 1 is given by

EðXnð1;pcÞÞ ¼2k�1ðuðnÞ þuðn=pÞÞ; pkn2k�1ð2uðnÞ þ ðpc � 2pþ 2Þuðn=pÞÞ; pckn; c P 2

2kðuðnÞ þ ðpc � pþ 1Þuðn=pÞÞ; pc,n:

8><>: ð9Þ

Proof. Let p = ps and c = cs, where 1 6 s 6 k. Let j ¼ pb11 pb2

2 � � � pbkk � J be a representation of an arbitrary index 0 6 j 6 n � 1,

where gcd(J,n) = 1. The jth eigenvalue of Xnð1; pcss Þ is given by

kj ¼ cðj;nÞ þ cðj;n=pcss Þ:

Suppose that there exists a prime number pijj for some i – s such bi 6 ai � 2. This implies that p2i jtn;j and p2

i jtn=pcss ;j. Further-

more, we have lðtn;jÞ ¼ lðtn=pcss ;jÞ ¼ 0 and thus kj = 0.

If bs 6 as � cs � 1 then p3s jtn;j and p2

s jtn=pcss ;j

. Similarly, we conclude that kj = 0.For an arbitrary index j, define the set P = {1 6 i 6 kji – s,bi = ai � 1}.Let Jl = {0 6 j 6 n � 1jbs = as � l,ai � 1 6 bi 6 ai for i – s}, for 0 6 l 6 cs + 1.

Case 1. For l = 0 and j 2 J0 we have

tn;j ¼n

gcdðj;nÞ ¼pa1

1 pa22 � � �pas

s � � �pakk

pb11 pb2

2 � � � pass � � �pbk

k

¼Y

i2Ppi:

On the other hand, it followscs a1 a2 as�cs ak Y
tn=pcs
s ;j ¼n=ps

gcdðj;n=pcss Þ¼ p1 p2 � � �ps � � �pk

pb11 pb2

2 � � �pas�css � � �pbk

k

¼i2P

pi:

The jth eigenvalue is given by cs cs

kj ¼ cðj;nÞ þ cðj;n=pcss Þ ¼ ð�1ÞjPj uðnÞ

uðQ

i2PpiÞþ ð�1ÞjPj uðn=ps Þ

uðQ

i2PpiÞ¼ ð�1ÞjPjuðnÞ þuðn=ps Þ

uðQ

i2PpiÞ:

The number of indices j 2 J0 with the same set P is equal to the number of J such that

gcd J;n

pb11 pb2

2 � � �pass � � �pbk

k

!¼ 1:

The last equation implies that the number of such indices is equal to the Euler’s totient function

un

pb11 pb2

2 � � �pass � � � pbk

k

!¼ u

Yi2P

pi

� �:

Case 2. Let l = 1 and for j 2 J1 we similarly obtain tn;j ¼ ps

Qi2Ppi and tn=pcs

s ;j ¼Q

i2Ppi. Therefore, the jth eigenvalue is given by

kj ¼ ð�1ÞjPjþ1 uðnÞuðps

Qi2PpiÞ

þ ð�1ÞjPj uðn=pcss Þ

uðQ

i2PpiÞ¼ ð�1ÞjPj ð�uðnÞ þ ðps � 1Þuðn=pcs

s ÞÞðps � 1Þuð

Qi2PpiÞ

:

The number of indices j 2 J1 with the same set P is equal to

un

pb11 pb2

2 � � �pas�1s � � �pbk

k

!¼ u ps

Yi2P

pi

� �¼ ðps � 1Þu

Yi2P

pi

� �:

Case 3. For 2 6 l 6 cs and j 2 Jl we obtain pas�cs�minðas�l;as�csÞs ktn=pcs

s ;j which implies that ps-tn=pcss ;j and tn=pcs

s ;j ¼Q

i2Ppi. Sincepl

sjtn;j and l P 2 it holds that l(tn,j) = 0. Therefore, the jth eigenvalue is given by

kj ¼ ð�1ÞjPj uðn=pcss Þ

uQ

i2Ppi

� � ¼ ð�1ÞjPj uðn=pcss Þ

uQ

i2Ppi

� � :
The number of indices j 2 Jl with the same set P is equal to


un

pb11 pb2

2 � � �pas�ls � � �pbk

k

!¼ u pl

s

Yi2P

pi

� �¼ pl�1

s ðps � 1ÞuY

i2Ppi

� �:

Case 4. For l = cs + 1 and j 2 Jcsþ1 we obtain pcsþ1s ktn;j and c(j,n) = l(tn,j) = 0. Also, it holds that pas�cs�minðas�cs�1;as�csÞ

s ktn=pcss ;j

which yields that psktn=pcss ;j. Therefore, the j-th eigenvalue is given by

kj ¼ ð�1ÞjPjþ1 uðn=pcss Þ

uðps

Qi2PpiÞ

¼ ð�1ÞjPjþ1 uðn=pcss Þ

uðps

Qi2PpiÞ

:

The number of indices j 2 Jl with the same set P is equal to

un

pb11 pb2

2 � � �pas�cs�1s � � �pbk

k

!¼ u pcsþ1

s

Yi2P

pi

� �¼ pcs

s ðps � 1ÞuY

i2Ppi

� �:

After all mention cases, the energy of Xnð1; pcss Þ is given by

EðXnð1;pcss ÞÞ ¼

Xn�1

j¼0

jkjj ¼X

P # f1;2;...;kgnfsg

uðnÞ þuðn=pcss Þ

uQ

i2Ppi

� � �uY

i2Ppi

� �þuðnÞ � ðps � 1Þuðn=pcs

s Þðps � 1Þu

Qi2Ppi

� � � ðps � 1ÞuY

i2Ppi

� �

þXcs

l¼2

uðn=pcss Þ

uQ

i2Ppi

� � � pl�1s ðps � 1Þu

Yi2P

pi

� �þ uðn=pcs

s Þðps � 1Þu

Qi2Ppi

� � � pcss ðps � 1Þu

Yi2P

pi

� �!: ð10Þ

If as = 1 then Jl = ; for l P 2 and cs = 1. Since the Euler totient function is multiplicative, for as = 1 we have u(n) = (ps � 1)u(n/ps). Thus, the relation (10) becomes

EðXnð1;psÞÞ ¼ 2k�1 � ðuðnÞ þuðn=psÞ þuðnÞ � ðps � 1Þuðn=psÞÞ ¼ 2k�1ðuðnÞ þuðn=psÞÞ:

If as = cs P 2 then Jcsþ1¼ ; since bs = as � cs � 1 < 0 is not defined. Thus, the relation (10) is reduced to the first three sum-

mands as followsc

!
EðXnð1;pcs
s ÞÞ ¼ 2k�1 � ðuðnÞ þuðn=psÞÞ þ ðuðnÞ � ðps � 1Þuðn=psÞÞ þ ðps � 1Þuðn=pcss ÞXs

l¼2

pl�1s

¼ 2k�1 � 2uðnÞ þ ðps � 2Þuðn=pcss Þ þ psðpcs�1

s � 1Þuðn=pcss Þ

� �¼ 2k�1ð2uðnÞ þ ðpcs

s � 2ps þ 2Þuðn=psÞÞ:

If as > cs P 2 the formula (10) is composed of four summands, thus we have

EðXnð1;pcss ÞÞ ¼ 2k�1ð2uðnÞ þ ðpcs

s � 2ps þ 2Þuðn=psÞ þ pcss uðn=pcs

s ÞÞ ¼ 2kðuðnÞ þ ðpcss � ps þ 1Þuðn=psÞÞ:


Theorem 4.2. Let n P 4 be an arbitrary integer. Then the energy of the integral circulant graph Xn(p,q) for p = ps and q = pt, where1 6 s < t 6 k, is given by

EðXnðp; qÞÞ ¼

2kuðnÞ; pkn qkn3 � 2k�1uðnÞ; 2kn q2jn2k�1ð2uðnÞ þuðn=ptÞuðptÞÞ; pkn q2jn p – 2

2k�1ð2uðnÞ þuðn=psÞuðpsÞÞ; p2jn qkn2k�1ð2uðnÞ þuðn=psÞuðpsÞ þuðn=ptÞuðptÞÞ; p2jn q2jn

8>>>>>>><>>>>>>>:

: ð11Þ

Proof. Let j ¼ pb11 pb2

2 � � � pbkk � J be a representation of an arbitrary index 0 6 j 6 n � 1, where gcd(J,n) = 1. The jth eigenvalue of

Xn (ps,pt) is given by

kj ¼ cðj;n=psÞ þ cðj;n=ptÞ:

Suppose that there exists prime number pijj for some i – s, t such bi 6 ai � 2. This implies that p2i jtn=ps ;j and p2

i jtn=pt ;j. Fur-thermore, we have lðtn=pt ;jÞ ¼ lðtn=ps ;jÞ ¼ 0 and thus kj = 0.

If bs 6 as � 3 then p3s jtn=pt ;j and p2

s jtn=ps ;j. Similarly, we conclude that kj = 0.


If bt 6 at � 3 then p3t jtn=ps ;j and p2

t jtn=pt ;j. Similarly, we conclude that kj = 0.

For an arbitrary index j, define the set P = {1 6 i 6 kji – s, t,bi = ai � 1}.

Let Jl1 ;l2 ¼ f0 6 j 6 n� 1 j bs ¼ as � l1; bt ¼ at � l2; ai � 1 6 bi 6 ai for i – s; tg.For j 2 Jl1 ;l2 , where 0 6 l1, l2 6 2, we have

tn=ps ;j ¼n=ps

gcdðj;n=psÞ¼ pas�1�minðas�1;as�l1Þ

s pat�ðat�l2Þt

Yi2P

pi ¼pl2

t

Qi2Ppi; 0 6 l1 6 1

pspl2t

Qi2Ppi; l1 ¼ 2

8><>: : ð12Þ

Similarly it follows

tn=pt ;j ¼pl1

sQ

i2Ppi; 0 6 l2 6 1

ptpl1sQ

i2Ppi; l2 ¼ 2

(: ð13Þ

The number of indices j 2 Jl1 ;l2 with the same set P is equal to the number of J such that

gcd J;n

pb11 pb2

2 � � �pas�l1s � � �pat�l2

t � � �pbkk

!¼ 1:

The last equation implies that the number of such indices is equal to

un

pb11 pb2

2 � � �pas�l1s � � �pat�l2

t � � �pbkk

!¼ u pl1

s pl2t

Yi2P

pi

� �: ð14Þ

Now, we distinguish four cases depending on the values of l1 and l2.

Case 1. 0 6 l1, l2 6 1. Q Q
According to the relations (12) and (13) it follows tn=ps ;j ¼ pl2
t i2Ppi and tn=pt ;j ¼ pl1s i2Ppi and therefore the jth eigenvalue is

given by

kj ¼ cðj;n=psÞ þ cðj;n=ptÞ ¼ ð�1ÞjPjþl2 uðn=psÞuðpl2

t

Qi2PpiÞ

þ ð�1ÞjPjþl1 uðn=ptÞuðpl1

sQ

i2PpiÞ

¼ ð�1ÞjPj ð�1Þl2uðn=psÞuðpl1s Þ þ ð�1Þl1uðn=ptÞuðp

l2t Þ

uðpl1s Þuðpl2

t ÞuQ

i2Ppi

� � : ð15Þ

If l1 = l2 then

jkjj ¼uðn=psÞuðp

l1s Þ þuðn=ptÞuðp

l2t Þ

uðpl1s Þuðpl2

t ÞuQ

i2Ppi

� � ;

while for ł1 – l2 we have

jkjj ¼ð�1Þl2uðn=psÞuðp

l1s Þ þ ð�1Þl1uðn=ptÞuðp

l2t Þ

uðpl1s Þuðpl2

t ÞuQ

i2Ppi

� � ;

except for ps = 2, p2t jn and n 2 4Nþ 2.

It can be noticed that the numerator of the above relation for l1 = 0 and l2 = 1 is reduced to

uðn=ptÞuðptÞ �uðn=psÞ; ð16Þ

while for l1 = 1 and l2 = 0 we have

uðn=psÞuðpsÞ �uðn=ptÞ: ð17Þ

Since Euler totient function is multiplicative, for as = 1 we have

uðnÞ ¼ ðps � 1Þuðn=psÞ:

Therefore, if pskn and ptkn the above expressions are equivalent to u(n) � u(n/ps) and u(n) � u(n/pt).Now assume that p2

s jn and p2t jn. We may conclude that both expressions (16) and (17) are greater than zero if and only if

(ps � 1)(pt � 1) > 1. The last relation is trivially satisfied.If p2

s jn and ptkn then expression (16) is equivalent to u(n) � u(n/ps), which is greater than zero. Expression 17 is greater orequal to zero if and only if (ps � 1)(pt � 2) P 1. This is true, since pt > ps P 2.

If pskn and p2t jn then expression (17) is equivalent to u(n) � u(n/pt), which is greater than zero. Expression (16) is greater


or equal to zero if and only if (ps � 2) (pt � 1) P 1. This is true, only if ps > 2. Therefore, for ps ¼ 2; p2t jn and n 2 4Nþ 2 we have

that

jkjj ¼uðnÞ �uðn=ptÞuðptÞuðpl1

s Þuðpl2t Þuð

Qi2PpiÞ

; ð18Þ

if l1 = 0 and l2 = 1, while

jkjj ¼uðnÞ �uðn=ptÞ

uðpl1s Þuðpl2

t ÞuQ

i2Ppi

� � ; ð19Þ

if l1 = 1 and l2 = 0.The number of indices j 2 Jl1 ;l2 with the same set P in all mentioned cases is given by (14) and equals

uðpl1s Þuðp

l2t Þu

Yi2P

pi

� �:

Case 2. l1 = 2, 0 6 l2 6 1. Q
According to relation (13) we have that tn=pt ;j ¼ p2
s i2Ppi, which further implies that c(j,n/pt) = 0. Now, using relation (12) itholds that tn=pt ;j ¼ psp

l2t

Qi2Ppi and thus

kj ¼ cðj;n=psÞ ¼ ð�1ÞjPjþl2þ1 uðn=psÞuðpsÞuðp

l2t Þu

Qi2Ppi

� � :
The number of indices j 2 Jl1 ;l2 with the same set P is given by (14) and equals
psuðpsÞuðpl2t Þu

Yi2P

pi

� �:

Case 3. 0 6 l1 6 1, l2 = 2.
In this case we obtain symmetric expressions for kj and the number of indices with given set P. Case 4. l1 = l2 = 2. Q Q
According to the relations (12) and (13) we have that tn=ps ;j ¼ p2t i2Ppi and tn=pt ;j ¼ p2

s i2Ppi, which further implies kj =c(j,n/ps) = c(j,n/pt) = 0.By summarizing all formulas in mention cases, the energy of Xn(ps, pt) is given by

EðXnðps;ptÞÞ ¼Xn�1

j¼0

jkjj ¼X

P # f1;2;...;kgnfs;tgðuðn=psÞ þuðn=ptÞÞ þ ðuðn=psÞuðpsÞ þuðn=ptÞuðptÞÞ þ ðuðn=ptÞuðptÞð

�uðn=psÞÞ þ ðuðn=psÞuðpsÞ �uðn=ptÞÞ þ 2uðn=psÞps þ 2uðn=ptÞptÞ: ð20Þ

If as = at = 1 then only nonempty sets are J0,0, J0,1, J1,0 and J1,1. Thus, the relation (20) becomes

EðXnðps;ptÞÞ ¼ 2k�2 � ððuðn=psÞ þuðn=ptÞÞ þ ðuðn=psÞuðpsÞ þuðn=ptÞuðptÞÞ þ ðuðn=ptÞuðptÞ �uðn=psÞÞþ ðuðn=psÞuðpsÞ �uðn=ptÞÞ

¼ 2k�2ð4uðnÞÞ ¼ 2kuðnÞ: ð21Þ

If as = 1, at > 1 and ps – 2 then J2,0, J2,1 and J2,2 are the empty sets. Also, for at > 1 we have

uðnÞ ¼ ðpt � 1Þpat�1t uðn=pat

t Þ ¼ ptuðpat�1t Þuðn=pat

t Þ ¼ ptuðn=ptÞ > ðpt � 1Þuðn=ptÞ:

Therefore, from the relation (20) follows

EðXnðps;ptÞÞ ¼ 2k�2 � 2ðuðnÞ þuðn=ptÞuðptÞÞ þ 2uðn=ptÞptð Þ ¼ 2k�1ð2uðnÞ þuðn=ptÞuðptÞÞ: ð22Þ

If as = 1, at > 1 and ps = 2, according to relations (18) and (19) the energy is equal to

EðXnðps;ptÞÞ ¼ 2k�2 � ðuðn=psÞ þuðn=ptÞÞ þ ðuðn=psÞuðpsÞ þuðn=ptÞuðptÞÞ þ ðuðnÞ �uðn=ptÞuðptÞÞ þ ðuðnÞ�uðn=ptÞÞ þ 2ptuðptÞ

¼ 2k�1ð2uðnÞ þuðn=ptÞptÞ ¼ 3 � 2k�1uðnÞ: ð23Þ
If as > 1 and at = 1, we have similar equation as in the previous case:
EðXnðps;ptÞÞ ¼ 2k�1ð2uðnÞ þuðn=psÞuðpsÞÞ:
If as > 1 and at > 1 then all sets Jl1 ;l2 , for 0 6 l1, l2 6 2 are nonempty and thus the energy is equal to


EðXnðps; ptÞÞ ¼ 2k�2 � 2ðuðn=psÞuðpsÞ þuðn=ptÞuðptÞÞ þ 2uðn=psÞps þ 2uðn=ptÞptð Þ

¼ 2k�1ð2uðnÞ þuðn=psÞps þuðn=ptÞuðptÞÞ: ð24Þ


5. Classes of non-cospectral graphs with equal energy

Let n ¼ p1p2 � � � pspasþ1sþ1 � � � p

akk be a prime factorization of n, where ai P 2 for s + 1 6 i 6 k. Using the result of Theorem 4.2 we

see that the energy of integral circulant graph Xn(pi,pj) does not depend on the choice of pi and pj, if pi, pjkn. Also, the sameconclusion can be derived if we consider the graphs Xn(2,pj) for aj P 2 and n 2 4Nþ 2.

Since the order of the graph Xn(pi,pj) is equal to u (n/pi) + u(n/pj), which is at the same time the largest eigenvalues also,we can construct at least s + 1 non-cospectral regular n-vertex hyperenergetic graphs,

Xnð1Þ;Xnðp1; p2Þ;Xnðp1; p3Þ; . . . ;Xnðp1;psÞ;

with equal energy. Similarly, we obtain the second class of k � s non-cospectral graphs with equal energy.

Xnð2;psþ1Þ;Xnð2;psþ2Þ; . . . ;Xnð2;pkÞ;

Moreover, we can consider a square-free number n = p1p2 � � �pk and prove that the following k2

� graphs

Xnðp1;p2Þ;Xnðp1;p3Þ; . . . ;Xnðpk�1;pkÞ;

are non-cospectral.Consider the integral circulant graph Xn(pi,pj). The largest eigenvalue and the degree of Xn(pi,pj) is u(n/pi) + u(n/pj).

According to the proof of Theorem 4.2 from [22], the second largest value among jk1j, jk2j, . . . , jkn�1j equals

sðXnðpi; pjÞÞ ¼maxu n

pi

� �þu n

pj

� �uðpÞ ;

uðnÞ �u npi

� �uðpjÞ

;uðnÞ �u n

pj

� �uðpiÞ

;2uðnÞuðpipjÞ

8<:

9=;

¼ un

pipj

!�max

pi þ pj � 2uðpijÞ

;pi � 2;pj � 2;2

( ); ð25Þ

where pij denotes the smallest prime number dividing npipj

.Assume that graphs Xn(pi,pj) and Xn(pr,pq) are cospectral, with pj > pi and pq > pr. Furthermore, assume that pi > pr.

Case 1. pi > 3 and pr > 3.
From pj > pi > 3 it easily follows that
sðXnðpi; pjÞÞ ¼ un

pipj

!� ðpj � 2Þ:

By equating the largest eigenvalues of these graphs and the values s (Xn(pi,pj)) and s(Xn(pr,pq)), it follows

uðprpqÞ � ðpi þ pj � 2Þ ¼ uðpipjÞ � ðpr þ pq � 2Þ; ð26ÞuðprpqÞ � ðpj � 2Þ ¼ uðpipjÞ � ðpq � 2Þ: ð27Þ

Notice that we used the multiplicative property of the Euler function.By subtraction, we get

ðpr � 1Þðpq � 1Þ � pi ¼ ðpi � 1Þðpj � 1Þ � pr : ð28Þ

Assume without loss of generality that pi < pr. It follows that pijpj � 1 and prjpq � 1. Since piju(pj) and prju(pq), from the rela-tion (26), we conclude that piju(prpq) and prj u(pipj). Since pi < pr, we have prjpj � 1 and from the relation (28) it holds thatp2

r jpq � 1. Similarly, from the relation (26) it follows that p2r jpj � 1 and again according to (26) p3

r jpq � 1 holds. Using infinitedescent, we get that four-tuple (pi,pj,pr,pq) does not exist.

Case 2. pi > 3 and pr = 3.
We distinguish two cases depending on the values of prq. Let prq = 2. Then, according to the relation (25) we have that !
sðXnðpr; pqÞÞ ¼ un

3pq� ðpq þ 1Þ:


uð3pqÞ � ðpi þ pj � 2Þ ¼ uðpipjÞ � ðpq þ 1Þuð3pqÞ � ðpj � 2Þ ¼ uðpipjÞ � ðpq þ 1Þ:


The last two equation hold only if pi + pj � 2 = pj � 2 which a contradiction.Let prq > 2. Since pq > pr = 3 and therefore pq P 5, we have that

pq � 2 Ppq þ 1

2P

pq þ 1uðprqÞ

:

From the last relation we conclude that


3pq

!�

pq þ 1uðprqÞ

:


uð3pqÞ � ðpi þ pj � 2Þ ¼ uðpipjÞ � ðpq þ 1Þuð3pqÞ �uðprqÞ � ðpj � 2Þ ¼ uðpipjÞ � ðpq þ 1Þ:

From the last relations we see that pi + pj � 2 = u(prq) � (pj � 2) holds. Next, it holds that pi 6 pj � 2, which further impliesu(prq) � (pj � 2) 6 2(pj � 2). But that is only the case if u(prq) 6 2 or equivalently prq 6 3, which is a contradiction.

Case 3. pi > 3 and pr = 2.
We distinguish two cases depending on the values of pq. Let pq = 3. From the relation (25) it can be concluded that
sðXnðpr; pqÞÞ ¼ 2 �u n6

� �:


uð6Þ � ðpi þ pj � 2Þ ¼ 3 �uðpipjÞ; ð29Þ

uð6Þ � ðpj � 2Þ ¼ 2 �uðpipjÞ:

By subtraction, we get

uð6Þ � pi ¼ uðpipjÞ ¼ ðpi � 1Þðpj � 1Þ:

From the last relation it holds that pijpj � 1 and combining with the relation (29) we obtain that pijpj � 2. This is a contra-diction, since pj � 2 and pj � 2 are relatively prime.Let pq > 3. Since the following inequality holds

pq � 2 Ppq

2P

pq

uðprqÞ;

we have


3pq

!� ðpq � 2Þ:

Now, this case is reduced to the Eqs. (26) and (27) from Case 1, where we obtained a contradiction.Case 4. pi = 3 and pr = 2.

Since pq – pi we have pq P 5, which further implies max{pq/u(prq),pq � 2,2)} = pq � 2. Therefore, it holds that


2pq

!ðpq � 2Þ:

Moreover, as pj – pr and pr = 2, we obtain pij = 2. Thus, we conclude

maxpj þ 1uðpijÞ

;1;pj � 2;2Þ( )

¼ pj þ 1

and

sðXnðpi; pjÞÞ ¼ un

3pj

!ðpj þ 1Þ:


uð2pqÞ � ðpj þ 1Þ ¼ uð3pjÞ � pq; ð30Þuð2pqÞ � ðpj þ 1Þ ¼ uð3pjÞ � ðpq � 2Þ: ð31Þ

From the previous relations we trivially get that four-tuple (pi,pj,pr,pq) does not exist in this case.


This way we actually prove that two cospectral integral circulant graphs ICGn(D1) and ICGn(D2) must be isomorphic i.e.D1 = D2, for a square-free number n and two-element divisor sets D1 and D2 containing prime divisors. Therefore, we supportconjecture proposed by So [34], that two graphs ICGn(D1) and ICGn(D2) are cospectral if and only if D1 = D2. The conjecturewas only proven for the trivial cases where n being square-free and product of two primes. Our result is obviously one formof generalization.

6. Concluding remarks

In this paper we focus on some global characteristics of the energy of integral circulant graphs such as energy modulo fourand existence of non-cospectral graphs classes with equal energy. We also find explicit formulas for the energy of ICGn(D)classes with two-element set D. In contrast to [22], the calculation of these formulas require extensive discussion in manydifferent cases. Some further generalizations on this topic would require much more case analysis. The examples of suchgeneralizations are calculating the energy of the graphs with three or more divisors, graphs with square-free orders etc.The general problem of calculating the energy of ICGn(D) graphs seems very difficult, since as we increase the number ofdivisors in D we have more sign changes in Ramanujan functions c(n, i).

For the further research we also propose some new general characteristics of the energy such as studying minimal andmaximal energies for a given integral circulant graph, and characterizing the extremal graphs. We will use the following niceresult from [18,37].

Theorem 6.1. Let G be a regular graph on n vertices of degree r > 0. Then

EðGÞP n;

with equality if and only if every component of G is isomorphic to the complete bipartite graph Kr,r.The proof is based on the estimation

EðGÞP M22ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M2M4p ;

where M2 = 2m and M4 are spectral moments of graph G, defined as

Mk ¼Xn

i¼1

kki :

The fourth moment is equal to M4 ¼ 8q� 2mþ 2P

v2V deg2ðvÞ, where q is the number of quadrangles in G.Let n be even number and assume that ICGn(D⁄) is isomorphic to Kn/2,n/2. The present authors in [6] proved the following

Theorem 6.2. Let d1,d2, . . . , dk be divisors of n such that the greatest common divisor gcd(d1,d2, . . . ,dk) equals d. Then the graphICGn(d1,d2, . . . , dk) has exactly d connected components isomorphic to ICGn=d

d1d ;

d2d ; . . . ; dk

d

� �.

In this case the complement of ICGn(D⁄), denoted by ICGnðDÞ, must contain exactly two connected components that are

cliques, and for D ¼ fd1; d2; . . . ; dkg we have gcd(d1,d2, . . . ,dk) = 2 and ICGn=2d12 ;

d22 ; . . . ; dk

2

� �is isomorphic to a complete graph

Kn/2. It simply follows that the set D must contain all even divisors of n and therefore D⁄ is the set of all odd divisors of n.Therefore, the degree of ICGn(D⁄) is equal to n

2 ¼P

d2D�u nd

� �and ICGn(D⁄) is isomorphic to a complete bipartite graph

Kn/2,n/2. Recall that the spectra of the complete bipartite graph Km,n consists offfiffiffiffiffiffiffimnp

, �ffiffiffiffiffiffiffimnp

and 0 with multiplicity n � 2.It follows that jkn=2j ¼ jk0j ¼ n

2 and for k – 0; n2 we have the following nice identity

kk ¼X

djn; d odd

cðk;dÞ ¼X

djn; d odd

l dgcdðk;dÞ

� � uðdÞu d

gcdðk;dÞ

� � ¼ 0:

Using computer search, for odd n the minimum is 2n 1� 1p

� �, where p is the smallest prime dividing n. The extremal inte-

gral circulant graph contains all divisors of n that are not divisible by p (and the complement of such graph is composed of pcliques). We leave this observation as a conjecture.

Acknowledgement

The authors gratefully acknowledge support from Research projects 174010, 174013 and 174033 of the Serbian Ministryof Science.

References

[1] A. Ahmadi, R. Belk, C. Tamon, C. Wendler, On mixing of continuous time quantum walks on some circulant graphs, Quant. Inform. Comput. 3 (2003)611–618.


[2] R.J. Angeles-Canul, R.M. Norton, M.C. Opperman, C.C. Paribello, M.C. Russell, C. Tamonk, Perfect state transfer integral ciculants and join of graphs,Quant. Inform. Comput. 10 (2010) 325–342.

[3] R.J. Angeles-Canul, R.M. Norton, M.C. Opperman, C.C. Paribello, M.C. Russell, C. Tamonk, Quantum perfect state transfer on weighted join graphs, Int. J.Quant. Inf. 7 (2009) 1429–1445.

[4] S. Akbari, F. Moazami, S. Zare, Kneser graphs and their complements are hyperenergetic, MATCH Commun. Math. Comput. Chem. 61 (2009) 361–368.[5] R.B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bull. Kerala Math. Assoc. 1 (2004) 129–132.[6] M. Bašic, A. Ilic, On the clique number of integral circulant graphs, Appl. Math. Lett. 22 (2009) 1406–1411.[7] M. Bašic, M. Petkovic, D. Stevanovic, Perfect state transfer in integral circulant graphs, Appl. Math. Lett. 22 (2009) 1117–1121.[8] M. Bašic, Characterization of circulant graphs having perfect state transfer, manuscript, 2010.[9] S. Blackburn, I. Shparlinski, On the average energy of circulant graphs, Linear Algebra Appl. 428 (2008) 1956–1963.

[10] A.S. Bonifácio, C.T.M. Vinagre, N.M.M. de Abreu, Constructing pairs of equienergetic and non-cospectral graphs, Appl. Math. Lett. 21 (2008) 338–341.[11] V. Brankov, D. Stevanovic, I. Gutman, Equienergetic chemical trees, J. Serb. Chem. Soc. 69 (2004) 549–553.[12] M. Christandl, N. Datta, T.C. Dorlas, A. Ekert, A. Kay, A.J. Landahl, Perfect transfer of arbitrary states in quantum spin networks, Phys. Rev. A 71 (2005)

032312.[13] P.J. Davis, Circulant matrices, Pure and Applied Mathematics, John Wiley & Sons, New York-Chichester-Brisbane, 1979.[14] C.D. Godsil, Periodic Graphs, arXiv:0806.2074v1 [math.CO], 12 June 2008.[15] I. Gutman, The energy of a graph, Ber. Math. Stat. Sekt. Forschungszent. Graz 103 (1978) 1–22.[16] I. Gutman, The Energy of a Graph: Old and New Results, in: A. Betten, A. Kohnert, R. Laue, A. Wassermann (Eds.), Algebraic Combinatorics and

Applications, Springer-Verlag, Berlin, 2001, pp. 196–211.[17] I. Gutman, Hyperenergetic molecular graphs, J. Serb. Chem. Soc. 64 (1999) 199–205.[18] I. Gutman, S.Z. Firoozabadi, J.A. de la Peña, J. Rada, On the energy of regular graphs, MATCH Commun. Math. Comput. Chem. 57 (2007) 435–442.[19] I. Gutman, X. Li, J. Zhang, Graph Energy, in: M. Dehmer, F. Emmert-Streib (Eds.), Analysis of Complex Networks, From Biology to Linguistics, Wiley-

VCH, Weinheim, 2009, pp. 145–174.[20] G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Oxford University Press, New York, 1980.[21] F.K. Hwang, A survey on multi-loop networks, Theor. Comput. Sci. 299 (2003) 107–121.[22] A. Ilic, The energy of unitary Cayley graphs, Linear Algebra Appl. 431 (2009) 1881–1889.[23] A. Ilic, M. Bašic, On the chromatic number of integral circulant graphs, Comput. Math. Appl. 60 (2010) 144–150.[24] A. Ilic, M. Bašic, I. Gutman, Triply equienergetic graphs, MATCH Commun. Math. Comput. Chem. 64 (2010) 189–200.[25] A. Ilic, Distance spectra and distance energy of integral circulant graphs, Linear Algebra Appl. 433 (2010) 1005–1014.[26] W. Klotz, T. Sander, Some properties of unitary Cayley graphs, Electron. J. Combin. 14 (2007) #R45.[27] J. Liu, B. Liu, E-L equienergetic graphs, MATCH Commun. Math. Comput. Chem. 66 (2011) 971–976.[28] O. Miljkovic, B. Furtula, S. Radenkovic, I. Gutman, Equienergetic and almost-equienergetic trees, MATCH Commun. Math. Comput. Chem. 61 (2009)

451–461.[29] S. Pirzada, I. Gutman, Energy of a graph is never the square root of an odd integer, Appl. Anal. Discrete Math. 2 (2008) 118–121.[30] H.S. Ramane, H.B. Walikar, Construction of equienergetic graphs, MATCH Commun. Math. Comput. Chem. 57 (2007) 203–210.[31] H.N. Ramaswamy, C.R. Veena, On the energy of unitary Cayley graphs, Electron. J. Combin. 16 (2007) #N24.[32] O. Rojo, L. Medina, Constructing graphs with energy

ffiffiffirp

EðGÞwhere G is a bipartite graph, MATCH Commun. Math. Comput. Chem. 62 (2009) 465–472.[33] N. Saxena, S. Severini, I. Shparlinski, Parameters of integral circulant graphs and periodic quantum dynamics, Int. J. Quant. Inf. 5 (2007) 417–430.[34] W. So, Integral circulant graphs, Discrete Math. 306 (2006) 153–158.[35] W. So, Remarks on some graphs with large number of edges, MATCH Commun. Math. Comput. Chem. 61 (2009) 351–359.[36] I. Stankovic, M. Miloševic, D. Stevanovic, Small and not so small equienergetic graphs, MATCH Commun. Math. Comput. Chem. 61 (2009) 443–450.[37] B. Zhou, I. Gutman, J.A. de la Peña, J. Rada, L. Mendoza, On spectral moments and energy of graphs, MATCH Commun. Math. Comput. Chem. 57 (2007)

183–191.[38] <http://en.wikipedia.org/wiki/Ramanujan’s_sum>.

http://en.wikipedia.org/wiki/Ramanujan's_sum

new results on the energy of integral circulant graphs

Documents