New results on the energy of integral circulant graphs

Applied Mathematics and Computation 218 (2011) 3470–3482

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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

Keywords: Integral circulant graphs Graph energy Eigenvalues Cospectral graphs

a b s t r a c t Circulant graphs are an important class of interconnection networks in parallel and distributed computing. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer as well. The integral circulant graph ICGn(D) has the vertex set Zn = {0, 1, 2, . . . , n  1} and vertices a and b are adjacent if gcd(a  b, n) 2 D, where D # {d : djn, 1 6 d < n}. These graphs are highly symmetric, have integral spectra and some remarkable properties connecting chemical graph theory and number theory. The energy of a graph was first defined by Gutman, as the sum of the absolute values of the eigenvalues of the adjacency matrix. Recently, there was a vast research for the pairs and families of non-cospectral graphs having equal energies. Following Bapat and 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 modulo 4. Furthermore, we establish some general closed form expressions for the energy of integral circulant graphs and generalize some results from Ilic´ [A. Ilic´, The energy of unitary Cayley graphs, Linear Algebra Appl. 431 (2009), 1881–1889]. We close the paper by proposing some open problems and characterizing extremal graphs with minimal energy among integral circulant graphs with n vertices, provided n is even. Ó 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 applications has grown during the last two decades, they appeared in coding theory, VLSI design, Ramsey theory and other areas. Recently there is vast research on the interconnection schemes based on circulant topology–circulant graphs represent an important class of interconnection networks in parallel and distributed computing (see [21]). Integral circulant graphs are also highly symmetric 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 quantum spin network based on a circulant topology to obtain good communication between them. In general, quantum spin system can be defined as a collection of qubits on a graph, whose dynamics is governed by a suitable Hamiltonian, without external control on the system. Different classes of graphs were examined for the purpose of perfect transferring the states of the systems. Since circulant graphs are mirror symmetric, they represent good candidates for the property of periodicity and thus integrality [12], which further implies that integral circulant graphs would be potential candidates for modeling the quantum spin networks that permit perfect state transfer [1–3,14,33]. These properties are primarily related to the spectra of these graphs. Indeed, the eigenvalues of the graphs are indexed in palindromic order (ki = kni) and can be represented by Ramanujan’s sums.

⇑ Corresponding author. E-mail addresses: [email protected] (A. Ilic´), [email protected] (M. Bašic´). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.08.094

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Bašic´ [7,8] established a condition under which integral circulant graphs have perfect state transfer and gave complete characterization these graphs. It turned out that the degree of 2 must be equal in a prime factorization of the difference of successive eigenvalues. Furthermore, exactly one of the divisors n/4 or n/2 have to belong to the divisor set D for any integral circulant graph ICGn(D) having perfect state transfer. In this paper we continue with studying parameters of integral circulant graphs like energy, having in mind application in chemical graph theory. We actually focus on characterization of 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 the diameter, bipartiteness and perfect state transfer. The present authors in [6,23] calculated the clique and chromatic number of 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 and eigenvalues of the unitary Cayley graphs. The latter group of authors proposed a generalization of unitary Cayley graphs named 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 is defined as the sum of absolute values of its eigenvalues [15,16,19]

EðGÞ ¼

n X

jki j:

i¼1

The concept of graph energy arose in chemistry where certain numerical quantities, such as the heat of formation of a hydrocarbon, 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 if E(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 maximality stable p-electron systems. In [22,31], the authors calculated the energy of unitary Cayley graphs and complement of unitary Cayley graphs, and establish the necessary and sufficient conditions for ICGn to be hyperenergetic. There was a vast research for 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 and Gutman [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 circulant graphs. In Section 3 we characterize the energy of integral circulant graph modulo 4, while in Section 4 we generalized formulas for the energy of integral circulant graphs from [22]. In Section 5, some larger families of graphs with equal energy are presented and further we support conjecture proposed by So [34], that two graphs ICGn(D1) and ICGn(D2) are cospectral if and only if D1 = D2. In concluding remarks we propose some open problems and characterize extremal graphs with minimal energy 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 n vertices, 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 S is 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 and only 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

Gn ðdÞ 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 positive divisors d of n, with d 6 n2. Theorem 2.1. A circulant graph G(n, S) is integral if and only if

S¼

[

Gn ðdÞ;

d2D

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 S1 = {s1js 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}jab1 2 S}. For a positive integer n > 1 the unitary

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Cayley graph Xn = Cay(Zn, Un) is defined by the additive group of the ring Zn of integers modulo n and the multiplicative group U n 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 the S Cayley set S = d2DGn(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 the other 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 a circulant matrix A. Define the polynomial Pn(z) by the entries of the first row of A,

Pn ðzÞ ¼

n1 X

ai  zi :

i¼0

The eigenvalues of A are given by

kj ¼ Pn ðxj Þ ¼

n1 X

ai  xji ;

0 6 j 6 n  1;

ð1Þ

i¼0

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 positive integer variables n and k defined by the formula

cðk; nÞ ¼

n X

n X

2p i

e n ak ¼

a¼1 gcdða;nÞ¼1

xak n ;

a¼1 gcdða;nÞ¼1

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

cðk; nÞ ¼ lðt n;k Þ 

uðnÞ n where t n;k ¼ gcdðk; nÞ uðtn;k Þ

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

kk ¼

X  n c k; ; d d2D

0 6 k 6 n  1:

ð2Þ

Using the well-known summation [20]

sðk; nÞ ¼

n1 X i¼0

8 > < 0 if xikn ¼ n if > :

n-k njk ;

we get that n1 X

cðk; nÞ ¼ 0:

ð3Þ

k¼0

For even n it follows n=21 X

n X

n=21 X

a¼1 gcdða;nÞ¼1

k¼0

cðk; nÞ ¼

k¼0

n=2 X

¼

xak n ¼

n=2 X

n=21 X

a¼1 gcdða;nÞ¼1

k¼0

an=2 xan þ n  xn

a¼1 gcdða;nÞ¼1

n1 X

! ðnaÞk xak n þ xn

! ¼ xak n

k¼0

n=2 X

ð1 þ 1Þ ¼ uðnÞ:

ð4Þ

a¼1 gcdða;nÞ¼1

Similarly, for odd n it follows ðn1Þ=2 X k¼0

cðk; nÞ ¼

uðnÞ 2

:

It also follows that if k  k0 (mod n) then c(k, n) = c(k0 , n). a Throughout the paper, we let n ¼ pa11 pa22    pk k , 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

ð5Þ

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n=d n ¼ gcdðn=d; iÞ gcdðn; idÞ

tn=d;i ¼ and

tn=d;ni ¼

n=d n ¼ : gcdðn=d; n  iÞ gcdðn; nd  idÞ

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

cði; n=dÞ ¼ lðtn=d;i Þ

uðn=dÞ uðn=dÞ ¼ lðtn=d;ni Þ ¼ cðn  i; n=dÞ; uðtn=d;i Þ uðtn=d;ni Þ

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 kni of ICGn(D) are equal. For i = 0 we have

k0 ¼

X

uðn=dÞ;

d2D

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

kn=2 ¼

X ð1Þd uðn=dÞ: d2D

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 2

ðn1Þ=2 X

jki j:

i¼1

Since x  jxj (mod 2), in order to characterize E(G) modulo 4 we consider the parity of the following sum ðn1Þ=2 X X EðGÞ X uðn=dÞ  þ cði; n=dÞðmod 2Þ: d2D 2 2 i¼1 d2D

Since n/d > 2, it follows that u(n/d) is even. After exchanging the order of the summation we have ðn1Þ=2 EðGÞ X uðn=dÞ X X  þ cði; n=dÞðmod 2Þ: d2D 2 2 i¼1 d2D

ð6Þ

By relation (3), we get that for every k it holds that kþn1 X

cði; nÞ ¼ 0:

ð7Þ

i¼k

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

Proof. Using the following relation

EðGÞ  2

0

n1 2

d1

¼ nd  d1 þ nd , the formula for graph energy (6) now becomes 2 2d n

1 ðlþ1Þd 2 X uðn=dÞ X B X þ cði; n=dÞ þ @ d2D 2 l¼0 i¼l n d2D

X

dþ1

Next we get

EðGÞ X uðn=dÞ  þ d2D 2 2

ðn=d1Þ=2 X i¼1

and using relation (5), we get that

cði; n=dÞðmod 2Þ

n1

2 X nðd1Þ i¼ 2d þ1

1

C cði; n=dÞAðmod 2Þ:

ð8Þ

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EðGÞ X uðn=dÞ uðn=dÞ  þ  uðn=dÞ  0ðmod 2Þ: d2D 2 2 2 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 jk0 j þ jkn=2 j þ 2

n=21 X

jki j:

i¼1

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

jk0 j þ jkn=2 j ¼

X d2D

   X  ð1Þd uðn=dÞ:   d2D

uðn=dÞ þ 

Also,

P 8 uðn=dÞ; if kn=2 > 0  > X uðn=dÞ X  < d2D; d ev en 1 u ðn=dÞ d   P ¼ þ  d2D ð1Þ S ¼  ðjk0 j þ jkn=2 jÞ ¼ : d2D 2 2 2  > uðn=dÞ; if kn=2 < 0 : d2D; d odd

If

n 2

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 2Þ

1; if

kn=2 > 0 and 4jn; or kn=2 < 0 and 4-n ðmod 2Þ

:

Therefore

X n=21 X EðGÞ Sþ cði; n=dÞ ðmod 2Þ: 2 d2D i¼1 Theorem 3.3. For even n, the energy of ICGn(D) is not divisible by four if and only if

n 2

R D and kn/2 is negative.

Proof. If d is even, we have 2n  1 ¼ 2d  nd  1. Since c(0, n/d) = u(n/d), it follows n=21 X

cði; n=dÞ ¼ cð0; n=dÞ þ

i¼1

If d is odd, we have  1 ¼

d1 2

cði; n=dÞ ¼ cð0; n=dÞ þ

i¼1

¼ uðn=dÞ þ For For

n 2

n 2

kn=d1 X

cði; n=dÞ ¼ uðn=dÞ þ

k¼1 i¼ðk1Þn=d n 2

n=21 X

d=2 X

n=d1 d X  cði; n=dÞ ¼ uðn=dÞ: 2 i¼0

 nd þ 12  nd  1. Similarly, using the relation (4), it follows ðd1Þ=2 X

kn=d1 X

k¼1

i¼ðk1Þn=d

cði; n=dÞ þ

n=21 X

cði; n=dÞ

i¼ððd1Þ=2Þn=d

n=ð2dÞ1 n=d1 X d1 X  cði; n=dÞ þ cði; n=dÞ ¼ uðn=dÞ þ uðn=dÞ ¼ 0: 2 i¼0 i¼0

R D, we have that S  0 (mod 2) and 4jE(G). 2 D, by combining above cases we have

X n=21 X d2D

cði; n=dÞ 

i¼1

1 þ ð1Þn=2 2

ðmod 2Þ:

For kn/2 > 0, it follows

EðGÞ 1 þ ð1Þn=2 1 þ ð1Þn=2 1 þ ð1Þn=2 Sþ  þ  0 ðmod 2Þ; 2 2 2 2 while for kn/2 < 0, we have

EðGÞ 1 þ ð1Þn=2 1  ð1Þn=2 1 þ ð1Þn=2 Sþ  þ  1 ðmod 2Þ: 2 2 2 2 This completes the proof. h

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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

8 k1 pkn > < 2 ðuðnÞ þ uðn=pÞÞ; EðX n ð1; pc ÞÞ ¼ 2k1 ð2uðnÞ þ ðpc  2p þ 2Þuðn=pÞÞ; pc kn; > : k 2 ðuðnÞ þ ðpc  p þ 1Þuðn=pÞÞ; pc ,n:

ð9Þ

cP2

b

Proof. Let p = ps and c = cs, where 1 6 s 6 k. Let j ¼ pb11 pb22    pkk  J be a representation of an arbitrary index 0 6 j 6 n  1, c where gcd(J, n) = 1. The jth eigenvalue of X n ð1; ps s Þ is given by

kj ¼ cðj; nÞ þ cðj; n=pcs s Þ: Suppose that there exists a prime number pijj for some i – s such bi 6 ai  2. This implies that p2i jt n;j and p2i jtn=pcs ;j . Furthers more, we have lðt n;j Þ ¼ lðtn=pscs ;j Þ ¼ 0 and thus kj = 0. 3 2 If bs 6 as  cs  1 then ps jt n;j and ps jt n=pcs ;j . Similarly, we conclude that kj = 0. s 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 a

tn;j ¼

pa1 pa2    pas s    pk k Y n ¼ 1b 2b ¼ p: i2P i gcdðj; nÞ p11 p22    pas s    pbkk

On the other hand, it cfollows a a 1 2 s

tn=pcs ;j ¼ s

a c

a

p1 p2    ps s s    pk k Y n=ps ¼ p: cs ¼ b1 b2 i2P i gcdðj; n=ps Þ p1 p2    pas s cs    pbkk

The jth eigenvalue is given by

kj ¼ cðj; nÞ þ cðj; n=pcs s Þ ¼ ð1ÞjPj

c

uðnÞ

c

uðn=ps s Þ uðnÞ þ uðn=ps s Þ Q Q þ ð1ÞjPj Q ¼ ð1ÞjPj : uð i2P pi Þ uð i2P pi Þ uð i2P pi Þ

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 pb22    pas s    pkk b

¼ 1:

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

u

!

n pb11 pb22

¼u

as

b    ps    pkk

Y

p i2P i

 :

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

kj ¼ ð1ÞjPjþ1

Q

i2P pi

and tn=pcs s ;j ¼

cs

Q

i2P pi .

Therefore, the jth eigenvalue is given by cs

uðnÞ uðn=ps Þ ðuðnÞ þ ðps  1Þuðn=ps ÞÞ Q Q þ ð1ÞjPj Q ¼ ð1ÞjPj : ðps  1Þuð i2P pi Þ uðps i2P pi Þ uð i2P pi Þ

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

u

!

n pb11 pb22    pas s 1    pkk b

 Y  Y  p : ¼ u ps i2P pi ¼ ðps  1Þu i i2P a c minða l;a c Þ

s s s Case 3. For 2 6 l 6 cs and j 2 Jl we obtain ps s s ktn=pcs ;j which implies that ps -t n=pcs ;j and tn=pcs ;j ¼ s s s l ps jt n;j and l P 2 it holds that l(tn,j) = 0. Therefore, the jth eigenvalue is given by

kj ¼ ð1ÞjPj

uðn=pcs s Þ uðn=pcs Þ Q  ¼ ð1ÞjPj Q s  : u i2P pi u i2P pi

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

Q

i2P pi .

Since

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u

!

n pb11 pb22

as l

   ps

b    pkk

 Y  Y  ¼ u pls i2P pi ¼ pl1 p : s ðps  1Þu i2P i c þ1

a cs minðas cs 1;as cs Þ

Case 4. For l = cs + 1 and j 2 J cs þ1 we obtain ps s kt n;j and c(j, n) = l(tn,j) = 0. Also, it holds that ps s which yields that ps kt n=pcs ;j . Therefore, the j-th eigenvalue is given by

kt n=pcs ;j s

s

kj ¼ ð1ÞjPjþ1

uðn=pcs s Þ uðn=pcs s Þ Q Q ¼ ð1ÞjPjþ1 : uðps i2P pi Þ uðps i2P pi Þ

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

u

!

n pb11 pb22

as cs 1

   ps

b    pkk

  Y  Y p : ¼ u pscs þ1 i2P pi ¼ pcs s ðps  1Þu i i2P c

After all mention cases, the energy of X n ð1; ps s Þ is given by

EðX n ð1; pcs s ÞÞ ¼

n1 X

X

jkj j ¼

j¼0

P # f1;2;...;kgnfsg

cs  uðnÞ  ðp  1Þuðn=pcs Þ Y  uðnÞþ uðn=p Þ Y s  s u Q  s  ðps  1Þu Q pi þ pi i2P i2P ðps  1Þu i2P pi u i2P pi

! cs Y  Y  X uðn=pcs s Þ l1 uðn=pcs s Þ cs Q   ps ðps  1Þu   Q :  p p ðp  1Þ u p þ þ s s i2P i i2P i ðps  1Þu i2P pi u i2P pi l¼2

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

EðX n ð1; ps ÞÞ ¼ 2k1  ðuðnÞ þ uðn=ps Þ þ uðnÞ  ðps  1Þuðn=ps ÞÞ ¼ 2k1 ðuðnÞ þ uðn=ps ÞÞ: If as = cs P 2 then J csþ1 ¼ ; since bs = as  cs  1 < 0 is not defined. Thus, the relation (10) is reduced to the first three summands as follows !

EðX n ð1; pcs s ÞÞ ¼ 2k1  ðuðnÞ þ uðn=ps ÞÞ þ ðuðnÞ  ðps  1Þuðn=ps ÞÞ þ ðps  1Þuðn=pcs s Þ

cs X

pl1 s

l¼2

  ¼ 2k1  2uðnÞ þ ðps  2Þuðn=pcs s Þ þ ps ðpcs s 1  1Þuðn=pcs s Þ ¼ 2k1 ð2uðnÞ þ ðpcs s  2ps þ 2Þuðn=ps ÞÞ: If as > cs P 2 the formula (10) is composed of four summands, thus we have

EðX n ð1; pcs s ÞÞ ¼ 2k1 ð2uðnÞ þ ðpcs s  2ps þ 2Þuðn=ps Þ þ pcs s uðn=pcs s ÞÞ ¼ 2k ðuðnÞ þ ðpcs s  ps þ 1Þuðn=ps ÞÞ: This completes the proof. h 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, where 1 6 s < t 6 k, is given by

8 k > 2 uðnÞ; > > > > k1 > > < 3  2 uðnÞ; EðX n ðp; qÞÞ ¼ 2k1 ð2uðnÞ þ uðn=pt Þuðpt ÞÞ; > > > > 2k1 ð2uðnÞ þ uðn=ps Þuðps ÞÞ; > > > : k1 2 ð2uðnÞ þ uðn=ps Þuðps Þ þ uðn=pt Þuðpt ÞÞ;

pkn qkn 2kn q2 jn pkn q2 jn p – 2 :

ð11Þ

2

p jn qkn p2 jn q2 jn

b

Proof. Let j ¼ pb11 pb22    pkk  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=ps Þ þ cðj; n=pt Þ: Suppose that there exists prime number pijj for some i – s, t such bi 6 ai  2. This implies that p2i jt n=ps ;j and p2i jtn=pt ;j . Furthermore, we have lðtn=pt ;j Þ ¼ lðt n=ps ;j Þ ¼ 0 and thus kj = 0. If bs 6 as  3 then p3s jt n=pt ;j and p2s jt n=ps ;j . Similarly, we conclude that kj = 0.

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If bt 6 at  3 then p3t jt n=ps ;j and p2t 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 J l1 ;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 J l1 ;l2 , where 0 6 l1, l2 6 2, we have

t n=ps ;j

8 l Q 2 0 6 l1 6 1 > < pt i2P pi ; Y n=ps as 1minðas 1;as l1 Þ at ðat l2 Þ l2 Q ¼ pt p ¼ ¼ ps : p p ; l1 ¼ 2 p i i2P > gcdðj; n=ps Þ : s t i2P i

ð12Þ

Similarly it follows

( t n=pt ;j ¼

pls1

Q

i2P pi ;

pt pls1

Q

0 6 l2 6 1

l2 ¼ 2 i2P pi ;

ð13Þ

:

The number of indices j 2 J l1 ;l2 with the same set P is equal to the number of J such that

gcd J;

!

n

¼ 1:

pb11 pb22    pas s l1    pat t l2    pkk b

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

u

!

n pb11 pb22

as l1

   ps

at l2

   pt

b    pkk

  Y ¼ u pls1 plt2 i2P 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 t n=ps ;j ¼ plt2 i2P pi and t n=pt ;j ¼ pls1 i2P pi and therefore the jth eigenvalue is given by

kj ¼ cðj; n=ps Þ þ cðj; n=pt Þ ¼ ð1ÞjPjþl2 ¼ ð1ÞjPj

uðn=ps Þ uðn=pt Þ þ ð1ÞjPjþl1 Q Q uðplt2 i2P pi Þ uðpls1 i2P pi Þ

ð1Þl2 uðn=ps Þuðpls1 Þ þ ð1Þl1 uðn=pt Þuðplt2 Þ : Q  uðpls1 Þuðplt2 Þu i2P pi

ð15Þ

If l1 = l2 then

jkj j ¼

uðn=ps Þuðpls1 Þ þ uðn=pt Þuðplt2 Þ ; Q  uðpls1 Þuðplt2 Þu i2P pi

while for ł1 – l2 we have

jkj j ¼

ð1Þl2 uðn=ps Þuðpls1 Þ þ ð1Þl1 uðn=pt Þuðplt2 Þ ; Q  uðpls1 Þuðplt2 Þu i2P pi

except for ps = 2, p2t jn and n 2 4N þ 2. It can be noticed that the numerator of the above relation for l1 = 0 and l2 = 1 is reduced to

uðn=pt Þuðpt Þ  uðn=ps Þ;

ð16Þ

while for l1 = 1 and l2 = 0 we have

uðn=ps Þuðps Þ  uðn=pt Þ:

ð17Þ

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

uðnÞ ¼ ðps  1Þuðn=ps Þ: 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 p2s 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 p2s jn and ptkn then expression (16) is equivalent to u(n)  u(n/ps), which is greater than zero. Expression 17 is greater or equal 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

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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 4N þ 2 we have that

jkj j ¼

uðnÞ  uðn=pt Þuðpt Þ ; Q uðpls1 Þuðplt2 Þuð i2P pi Þ

ð18Þ

if l1 = 0 and l2 = 1, while

jkj j ¼

uðnÞ  uðn=pt Þ Q ; uðpls1 Þuðplt2 Þu i2P pi

ð19Þ

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

uðpls1 Þuðplt2 Þu

Y

 p : i i2P

Case 2. l1 = 2, 0 6 l2 6 1. Q According to relation (13) we have that t n=pt ;j ¼ p2s i2P pi , which further implies that c(j, n/pt) = 0. Now, using relation (12) it l2 Q holds that tn=pt ;j ¼ ps pt i2P pi and thus

kj ¼ cðj; n=ps Þ ¼ ð1ÞjPjþl2 þ1

uðn=ps Þ Q : uðps Þuðplt2 Þu i2P pi

The number of indices j 2 J l1 ;l2 with the same set P is given by (14) and equals

ps uðps Þuðplt2 Þu

Y

p i2P i

 :

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 t n=ps ;j ¼ p2t i2P pi and tn=pt ;j ¼ p2s i2P pi , 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

EðX n ðps ; pt ÞÞ ¼

n1 X j¼0

jkj j ¼

X

ððuðn=ps Þ þ uðn=pt ÞÞ þ ðuðn=ps Þuðps Þ þ uðn=pt Þuðpt ÞÞ þ ðuðn=pt Þuðpt Þ

P # f1;2;...;kgnfs;tg

 uðn=ps ÞÞ þ ðuðn=ps Þuðps Þ  uðn=pt ÞÞ þ 2uðn=ps Þps þ 2uðn=pt Þpt Þ:

ð20Þ

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

EðX n ðps ; pt ÞÞ ¼ 2k2  ððuðn=ps Þ þ uðn=pt ÞÞ þ ðuðn=ps Þuðps Þ þ uðn=pt Þuðpt ÞÞ þ ðuðn=pt Þuðpt Þ  uðn=ps ÞÞ þ ðuðn=ps Þuðps Þ  uðn=pt ÞÞ ¼ 2k2 ð4uðnÞÞ ¼ 2k uðnÞ:

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  1Þptat 1 uðn=pat t Þ ¼ pt uðpat t 1 Þuðn=pat t Þ ¼ pt uðn=pt Þ > ðpt  1Þuðn=pt Þ: Therefore, from the relation (20) follows

EðX n ðps ; pt ÞÞ ¼ 2k2  ð2ðuðnÞ þ uðn=pt Þuðpt ÞÞ þ 2uðn=pt Þpt Þ ¼ 2k1 ð2uðnÞ þ uðn=pt Þuðpt ÞÞ:

ð22Þ

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

EðX n ðps ; pt ÞÞ ¼ 2k2  ðuðn=ps Þ þ uðn=pt ÞÞ þ ðuðn=ps Þuðps Þ þ uðn=pt Þuðpt ÞÞ þ ðuðnÞ  uðn=pt Þuðpt ÞÞ þ ðuðnÞ  uðn=pt ÞÞ þ 2pt uðpt Þ ¼ 2k1 ð2uðnÞ þ uðn=pt Þpt Þ ¼ 3  2k1 uðnÞ: If as > 1 and at = 1, we have similar equation as in the previous case:

EðX n ðps ; pt ÞÞ ¼ 2k1 ð2uðnÞ þ uðn=ps Þuðps ÞÞ: If as > 1 and at > 1 then all sets J l1 ;l2 , for 0 6 l1, l2 6 2 are nonempty and thus the energy is equal to

ð23Þ

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EðX n ðps ; pt ÞÞ ¼ 2k2  ð2ðuðn=ps Þuðps Þ þ uðn=pt Þuðpt ÞÞ þ 2uðn=ps Þps þ 2uðn=pt Þpt Þ ¼ 2k1 ð2uðnÞ þ uðn=ps Þps þ uðn=pt Þuðpt ÞÞ:

ð24Þ

This completes the proof. h 5. Classes of non-cospectral graphs with equal energy a

a

sþ1 Let n ¼ p1 p2    ps psþ1    pk k 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 same conclusion can be derived if we consider the graphs Xn(2, pj) for aj P 2 and n 2 4N þ 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,

X n ð1Þ; X n ðp1 ; p2 Þ; X n ðp1 ; p3 Þ; . . . ; X n ðp1 ; ps Þ; with equal energy. Similarly, we obtain the second class of k  s non-cospectral graphs with equal energy.

X n ð2; psþ1 Þ; X n ð2; psþ2 Þ; . . . ; X n ð2; pk Þ; Moreover, we can consider a square-free number n = p1p2    pk and prove that the following

 k graphs 2

X n ðp1 ; p2 Þ; X n ðp1 ; p3 Þ; . . . ; X n ðpk1 ; pk Þ; 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, . . . , jkn1j equals

      8   9
ð25Þ

where pij denotes the smallest prime number dividing pinpj . 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

n sðX n ðpi ; pj ÞÞ ¼ u pi pj

!

 ðpj  2Þ:

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

uðpr pq Þ  ðpi þ pj  2Þ ¼ uðpi pj Þ  ðpr þ pq  2Þ; uðpr pq Þ  ðpj  2Þ ¼ uðpi pj Þ  ðpq  2Þ:

ð26Þ 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 relation (26), we conclude that piju(prpq) and prj u(pipj). Since pi < pr, we have prjpj  1 and from the relation (28) it holds that p2r jpq  1. Similarly, from the relation (26) it follows that p2r jpj  1 and again according to (26) p3r jpq  1 holds. Using infinite descent, 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

sðX n ðpr ; pq ÞÞ ¼ u

n 3pq

!

 ðpq þ 1Þ:

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

uð3pq Þ  ðpi þ pj  2Þ ¼ uðpi pj Þ  ðpq þ 1Þ uð3pq Þ  ðpj  2Þ ¼ uðpi pj Þ  ðpq þ 1Þ:

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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 P

pq þ 1 pq þ 1 P : 2 uðprq Þ

From the last relation we conclude that

sðX n ðpr ; pq ÞÞ ¼ u

n 3pq

!



pq þ 1

uðprq Þ

:

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

uð3pq Þ  ðpi þ pj  2Þ ¼ uðpi pj Þ  ðpq þ 1Þ uð3pq Þ  uðprq Þ  ðpj  2Þ ¼ uðpi pj Þ  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 implies u(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

sðX n ðpr ; pq ÞÞ ¼ 2  u

n : 6

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

uð6Þ  ðpi þ pj  2Þ ¼ 3  uðpi pj Þ;

ð29Þ

uð6Þ  ðpj  2Þ ¼ 2  uðpi pj Þ: By subtraction, we get

uð6Þ  pi ¼ uðpi pj Þ 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 contradiction, since pj  2 and pj  2 are relatively prime. Let pq > 3. Since the following inequality holds

pq  2 P

pq pq P ; 2 uðprq Þ

we have

n sðX n ðpr ; pq ÞÞ ¼ u 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

sðX n ðpr ; pq ÞÞ ¼ u

! n ðpq  2Þ: 2pq

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

( ) pj þ 1 max ; 1; pj  2; 2Þ ¼ pj þ 1 uðpij Þ

and

! n ðpj þ 1Þ: sðX n ðpi ; pj ÞÞ ¼ u 3pj By equating the largest eigenvalues of these graphs and the values s (Xn(pi, pj)) and s(Xn(pr, pq)), it follows

uð2pq Þ  ðpj þ 1Þ ¼ uð3pj Þ  pq ; uð2pq Þ  ðpj þ 1Þ ¼ uð3pj Þ  ðpq  2Þ: From the previous relations we trivially get that four-tuple (pi, pj, pr, pq) does not exist in this case.

ð30Þ ð31Þ

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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 support conjecture proposed by So [34], that two graphs ICGn(D1) and ICGn(D2) are cospectral if and only if D1 = D2. The conjecture was only proven for the trivial cases where n being square-free and product of two primes. Our result is obviously one form of generalization. 6. Concluding remarks In this paper we focus on some global characteristics of the energy of integral circulant graphs such as energy modulo four and 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 many different cases. Some further generalizations on this topic would require much more case analysis. The examples of such generalizations 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 of divisors 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 and maximal energies for a given integral circulant graph, and characterizing the extremal graphs. We will use the following nice result from [18,37]. Theorem 6.1. Let G be a regular graph on n vertices of degree r > 0. Then

EðGÞ P 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

M 22 ffi; EðGÞ P pffiffiffiffiffiffiffiffiffiffiffiffiffi M2 M4 where M2 = 2m and M4 are spectral moments of graph G, defined as

Mk ¼

n X

kki :

i¼1

P 2 The fourth moment is equal to M 4 ¼ 8q  2m þ 2 v 2V deg ðv Þ, 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(d  1, d2, . . . , dk) equals d. Then the graph ICGn(d1, d2, . . . , dk) has exactly d connected components isomorphic to ICGn=d dd1 ; dd2 ; . . . ; ddk . In this case the complement of ICGn(D⁄), denoted by ICGn ðDÞ, must contain exactly two connected components that are   cliques, and for D ¼ fd1 ; d2 ; . . . ; dk g we have gcd(d1, d2, . . . , dk) = 2 and ICGn=2 d21 ; d22 ; . . . ; d2k 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.  P Therefore, the degree of ICGn(D⁄) is equal to n2 ¼ d2D u nd and ICGn(D⁄) is isomorphic to a complete bipartite graph pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi Kn/2,n/2. Recall that the spectra of the complete bipartite graph Km,n consists of mn,  mn and 0 with multiplicity n  2. n n It follows that jkn=2 j ¼ jk0 j ¼ 2 and for k – 0; 2 we have the following nice identity

X

kk ¼

cðk; dÞ ¼

djn; d odd

X djn; d odd



l

d uðdÞ    ¼ 0: d gcdðk; dÞ u gcdðk;dÞ

  Using computer search, for odd n the minimum is 2n 1  1p , where p is the smallest prime dividing n. The extremal integral circulant graph contains all divisors of n that are not divisible by p (and the complement of such graph is composed of p cliques). We leave this observation as a conjecture. Acknowledgement The authors gratefully acknowledge support from Research projects 174010, 174013 and 174033 of the Serbian Ministry of 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.

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