ON INTEGRAL CAYLEY SUM GRAPHS Marzieh Amooshahi and Bijan Taeri

Indian J. Pure Appl. Math., 47(4): 583-601, December 2016 °c Indian National Science Academy DOI: 10.1007/s13226-016-0204-5

Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran e-mails: [email protected]; [email protected] (Received 8 October 2014; after ﬁnal revision 18 March 2015; accepted 15 February 2016)

Let S be a subset of a ﬁnite abelian group G. The Cayley sum graph Cay+(G, S) of G with respect to S is a graph whose vertex set is G and two vertices g and h are joined by an edge if and only if g + h ∈ S. We call a ﬁnite abelian group G a Cayley sum integral group if for every subset S of G, Cay+(G, S) is integral i.e., all eigenvalues of its adjacency matrix are integers. n In this paper, we prove that all Cayley sum integral groups are represented by Z3 and Z2 , n ≥ 1,

where Zk is the group of integers modulo k. Also, we classify simple connected cubic integral Cayley sum graphs.

Key words : Cayley sum graph; integral graph; Cayley sum integral group.

1. INTRODUCTION

Let G be a ﬁnite abelian group and S be a subset of G. The Cayley sum graph Cay+(G, S) is the graph having the vertex set G and the edge set {{g, h} | g, h ∈ G, g + h ∈ S}. If S is a multi-set, then Cay+(G, S) contains multiple edges, and if there exists g ∈ G with 2g ∈ S, then the edge {g, g} is a semi-edge. A semi-edge of a graph is an edge with one endpoint. Unlike a loop, a semi-edge contributes just one to both the valency of its endpoint and the corresponding diagonal entry of the adjacency matrix. With this convention, Cay+(G, S) is a regular graph with valency |S|. A graph with no multiple edge and semi-edge is called simple.

Cayley sum graphs also are known under names addition Cayley graphs [12, 15, 18], addition graphs [8] and sum graphs [9]. While regular Cayley graphs are widely studied and there is a great expanse of literature concerning these, Cayley sum graphs have been largely disregarded or over- looked and it is possible to give a nearly conclusive list of all the literature on Cayley sum graphs; in 584 MARZIEH AMOOSHAHI AND BIJAN TAERI fact, [3] (independence number), [8] and [15] (hamiltonicity), [9] (expander properties), [11] (clique number), [12] (connectivity) and [4] (domination and total domination numbers) is a nearly complete list of papers, known to us, where Cayley sum graphs are addressed. To some extent, this situation may be explained by the fact that Cayley sum graphs are rather difficult to study. For instance, it is well-known and easy to prove that any connected Cayley graph on a finite abelian group with at least three elements is hamiltonian; however, apart from the results of [8], nothing seems to be known on hamiltonicity of Cayley sum graphs on finite abelian groups. Similarly, the connectivity of a Cayley graph on a finite abelian group is easy to determine, while determining the connectivity of a Cayley sum graph is a non-trivial problem, see [12].

A graph is called integral if it has an integral spectrum, that is all eigenvalues of its adjacency matrix are integers. The totally disconnected graph on n vertices and the complete graph on n vertices are examples of integral graphs with spectrum (0,..., 0) and (n − 1, −1,..., −1), respectively. The notion of integral graphs dates back to Harary and Schwenk [13] when they asked which graphs have integral spectra? In general, the answer of this question appears to be difﬁcult. However, by focusing on an speciﬁc class of graphs, it may become easier. For example, the path on n vertices is integral if and only if n = 2, also the cycle on n vertices is integral if and only if n = 3, 4, 6, see [6]. It has recently been discovered that integral graphs may be of interest for designing the network topology of perfect state transfer networks, see [2, 16].

It is interesting to know that among all connected cubic graphs, there are exactly 13 integral graphs [7]. In [1], Abdollahi and Vatandoost determined all connected cubic integral Cayley graphs. They proved that there are exactly seven connected cubic integral Cayley graphs. Considering a similar problem, in [5] the authors proved that there is exactly one simple connected cubic integral Cayley sum graph over non abelian groups. In this paper, we show that there are exactly six simple connected cubic integral Cayley sum graphs over abelian groups.

A ﬁnite abelian group G is called Cayley sum integral if for every subset S of G, Cay+(G, S) is integral. We show that all non-trivial abelian Cayley sum integral groups are represented by Z3 and n Z2 , n ≥ 1, where Zk is the group of integers modulo k.

2. SOME ELEMENTARY PROPERTIES OF CAYLEY SUM GRAPHS

Throughout the paper we assume that G is an additive ﬁnite abelian group. The order of G is denoted by |G|, and the order of an element g ∈ G, is denoted by ord(g). For a subset S of G, let S − S = + {s1 − s2 | s1, s2 ∈ S}. If g and h are adjacent in Cay (G, S), i.e. g + h ∈ S, we write g ∼ h. Also, N(g) = {h ∈ G | g + h ∈ S} denotes the set of neighbours of g. ON INTEGRAL CAYLEY SUM GRAPHS 585

In this section, we prove some elementary properties of Cayley sum graphs. Also, we determine the connected components of Cay+(G, S), where S ⊆ G. First we recall a necessary and sufﬁcient condition for a Cayley sum graph to be connected, proved by Lev [15].

Lemma 2.1 [15] — Let S be a subset of the ﬁnite abelian group G. In order for Cay+(G, S) to be connected it is necessary and sufﬁcient that S is not contained in a coset of a proper subgroup of G, except, perhaps, for the non-zero coset of a subgroup of index 2.

It follows from Lemma 2.1 that if Cay+(G, S) is connected and S is contained in some non-zero coset of a proper subgroup H of G, then |G : H| = 2 and S is not contained in any other coset.

In [8], Cheyne et al. proved that if S is a square free subset of G, then Cay+(G, S) is connected if and only if G = hSi and |G : hS − Si| ≤ 2. In [5], using Lemma 2.1, it is proved that the condition on S is superﬂuous.

Note that for a subset S of G we have S ⊆ hS−Si+s, for all s ∈ S, as t = (t−s)+s ∈ hS−Si+s, for every t ∈ S. Also, for every s, t ∈ S, we have hS − Si + s = hS − Si + t, as s − t ∈ hS − Si.

Lemma 2.2 [5] — Let S be a subset of G. Then Cay+(G, S) is connected if and only if G = hSi and |G : hS − Si| ≤ 2.

In the following proposition, we show that the only cycle of odd order as a Cayley sum graph is

C3, the cycle on 3 vertices. Also, every cycle of even order can be considered as a Cayley sum graph.

Proposition 2.3 — Let G be a non-trivial ﬁnite cyclic group and S ⊆ G. Then, Cay+(G, S) is a + + + cycle if and only if Cay (G, S) is isomorphic to Cay (Z3, Z3), Cay (Z4, {0, 1, 2}) or + Cay (Z2n, {1, 2n − 1}), where n ≥ 3.

+ + PROOF : First note that Cay (Z3, Z3) and Cay (Z4, {0, 1, 2}) are cycles of length 3 and 4, respectively, where each of their vertices is adjacent to a semi-edge. Also, the Cayley sum graph + + Cay (Z2n, {1, 2n − 1}), n ≥ 3, is a simple cycle of length 2n, since Cay (Z2n, {1, 2n − 1}) is connected and for any vertex u ∈ Z2n, we have N(u) = {1 − u, 2n − 1 − u}.

To prove the converse, suppose that G = Cay+(G, S) is a cycle. We consider two cases;

Case 1 : |G| is odd. Then S ⊆ G = 2G. Thus for any s ∈ S, there exists g ∈ G such that s = g + g ∈ S and so {g, g} is a semi-edge. Since G is a cycle we conclude that every vertex is adjacent to a semi-edge. Thus |S| = 3 and 2G ⊆ S. Therefore G = S and also |G| = |S| = 3.

Case 2 : |G| = 2n is even. If |S| = 3, then since G is a cycle, every vertex is adjacent to a semi-edge and so 2G ⊆ S. Therefore |G| = 4 or 6, since |2G| = n. If |G| = 4, then S = {0, 1, 2} or 586 MARZIEH AMOOSHAHI AND BIJAN TAERI

+ ∼ + {0, 2, 3}. It is easy to see that Cay (Z4, {0, 1, 2}) = Cay (Z4, {0, 2, 3}) is a cycle that every vertex + is adjacent to a semi-edge. If |G| = 6, then S = {0, 2, 4}. Lemma 2.2 implies that Cay (Z6,S) is not connected. So suppose that |S| = 2. Since G is a cycle we conclude that it has no semi-edges. + Now, since Cay (Z2n, {1, 2n − 1}) is a simple cycle and all simple cycles with the same length are isomorphic, the result holds.

Proposition 2.4 — Let G be a non-trivial ﬁnite cyclic group and S ⊆ G. Then, Cay+(G, S) is a + ∼ + path if and only if Cay (G, S) = Cay (Zn, {0, 1}), where n ≥ 2.

+ n PROOF : Put G = Cay (Zn, {0, 1}). If n is even, then vertices 0 and 2 of G are adjacent to semi- n n edges {0, 0} and { 2 , 2 }, respectively; and the other vertices are not adjacent to any semi-edges. If n is n+1 n+1 n+1 odd, then two vertices 0 and 2 of G are adjacent to semi-edges {0, 0} and { 2 , 2 }, respectively; and the other vertices are not adjacent to any semi-edges. Also, for any n, G is connected, by Lemma 2.2. Therefore, G is a path. The converse follows from the fact that all paths with the same length are isomorphic. 2

In the following theorem, the components of Cayley sum graphs are determined. We denote by C(x) the connected component containing a vertex x.

Theorem 2.5 — Let S be a subset of G and put H := hS − Si. If x is an element of G with 2x∈ / H + s, for some s ∈ S, then C(x) = (H + x) ∪ (H + s − x) is a bipartite subgraph of Cay+(G, S); while if 2x ∈ H + s, then C(x) = H + x. Moreover, for any y ∈ G with 2y∈ / H + s, the components C(x) and C(y) are isomorphic. Also, Cay+(G, S) has

1 (|G/H| + |A|) 2 connected components, where A = {H + x | 2x ∈ H + s, for some s ∈ S}.

PROOF : We have S ⊆ H + s. If s0 ∈ S ∩ H, then for all t ∈ S we have t = (t − s0) + s0 ∈ H and so S ⊆ H. Therefore either S ⊆ H or S ∩ H = ∅. We consider two below cases.

Case 1 : S ⊆ H. We claim that C(x) = (H + x) ∪ (H − x). For every h ∈ H, there exist n ∈ N Pn and s1, t1, s2, t2, . . . , sn, tn ∈ S such that h = i=1(si − ti). So we have

Xn x ∼ tn − x ∼ sn − tn + x ∼ tn−1 − sn + tn − x ∼ · · · ∼ t1 − (si − ti) − x ∼ h + x, i=2 a path between x and h + x. Then, H + x ⊆ C(x). Also, since S ⊆ H, there exist m ∈ N and ON INTEGRAL CAYLEY SUM GRAPHS 587

Pm r1, w1, r2, w2, . . . , rm, wm ∈ S such that tn = j=1(rj − wj). Thus we have

x ∼ wm − x ∼ rm − wm + x ∼ wm−1 − rm + wm − x ∼ · · · Xm ∼ w1 − (rj − wj) − x ∼ tn + x ∼ sn − tn − x ∼ tn−1 − sn + tn + x ∼ · · · j=2 Xn ∼ t1 − (si − ti) + x ∼ h − x, i=2 a path between x and h − x and so H − x ⊆ C(x). Therefore, (H + x) ∪ (H − x) ⊆ C(x).

To prove the reverse inclusion, let g ∈ C(x). Then, there exists a path from x to g, say x ∼ x1 ∼ x2 ∼ · · · ∼ xk ∼ g, for some x1, x2, . . . , xk ∈ G. So there exist s, s1, s2, . . . , sk ∈ S i−1 i such that xi = si − si−1 + ··· + (−1) s1 + (−1) x ∈ (H + x) ∪ (H − x), for every 1 ≤ i ≤ k and g = s − xk ∈ (H + x) ∪ (H − x). Thus, C(x) ⊆ (H + x) ∪ (H − x), as desired. 2

Note that if 2x ∈ H, then since H + x = H − x, we have C(x) = H + x; and if 2x∈ / H, then C(x) is a simple bipartite subgraph with partition sets H + x and H − x. Now for every x, y ∈ G with 2x, 2y∈ / H, the mapping ϕ : C(x) −→ C(y) given by ( h + x 7→ h + y; h − x 7→ h − y, is a bijection. Also, ϕ is adjacency preserving, as for all h1, h2 ∈ H, we have

h1 + x ∼ h2 − x ⇐⇒ h1 + x + h2 − x = h1 + h2 = s ∈ S

⇐⇒ s = h1 + h2 = h1 + y + h2 − y = ϕ(h1 + x) + ϕ(h2 − x)

⇐⇒ ϕ(h1 + x) ∼ ϕ(h2 − x).

Therefore, C(x) =∼ C(y).

Case 2 : S ∩H = ∅. By a similar method used in Case 1, we have C(x) = (H +x)∪(H +s−x), for some s ∈ S. If 2x ∈ H + s, then H + x = H + s − x. Therefore C(x) = H + x. If 2x∈ / H + s, then C(x) is a simple bipartite subgraph with partition sets H + x and H + s − x. For every x, y ∈ G such that 2x, 2y∈ / H + s, the mapping ϕ : C(x) −→ C(y) given by ( h + x 7→ h + y; h + s − x 7→ h + s − y, is a bijection. It is easy to see that ϕ preserves adjacency. 588 MARZIEH AMOOSHAHI AND BIJAN TAERI

Now, since Ã !   [ [ [ G = C(x) = x + H ∪  ((x + H) ∪ (s − x + H)), x∈G x+H∈A x+H/∈A we conclude that the number of components of Cay+(G, S) is 1 1 |A| + (|G/H| − |A|) = (|G/H| + |A|).2 2 2

+ 1 Theorem 2.5 states that for every subset S of G, the Cayley sum graph Cay (G, S) has 2 (|G/H|− |A|) isomorphic connected components of order 2|H|. Also, in Case 1, A is the set of all involutions of G/H, i.e. A = Inv(G/H) = {H + x | 2x ∈ H}.

Example 2.6 : Using the notation of Theorem 2.5, suppose that S ⊆ H and G/H = hg + Hi is a cyclic group of order n, where g ∈ G. According to the parity of n, we consider two cases.

Case 1 : n is odd. Then Inv(G/H) = {H} and thus there is only one component of order |H| and the order of each other components is 2|H|. If x ∈ H, then C(x) = H is a subgraph of order n−1 |H|. If x∈ / H, then C(x) = (H + x) ∪ (H − x). Thus, for every i = 1, 2,..., 2 , ig ∈ H and so C(ig) = (H + ig) ∪ (H + (n − i)g) is a bipartite subgraph of order 2|H|. In particular, if S ≤ G, then S = H and thus C(0) is a complete subgraph isomorphic to K|H|, the complete graph with |G| vertices, and the other components are complete bipartite subgraphs isomorphic to K|H|,|H|.

n n Case 2 : n is even. Then Inv(G/H) = {H, 2 g +H}. So C(0) and C( 2 g) are subgraphs of order n |H| and for every i = 1, 2,..., 2 − 1, C(ig) = (H + ig) ∪ (H + (n − i)g) is a bipartite subgraph of order 2|H|. In particular, if S ≤ G, then S = H and thus C(ig) is a complete bipartite subgraph n n isomorphic to K|H|,|H|, for every i = 1, 2,..., 2 − 1. Also, C(0) and C( 2 g) are complete subgraphs isomorphic to K|H|.

Eexample 2.7 : Using the notation of Theorem 2.5, suppose that S ⊆ H and G/H is an elementary abelian p−group, where p is prime. If p = 2, then there is no bipartite subgraph of order 2|H| and Cay+(G, S) is the union of subgraphs of order |H|. In particular, if S ≤ G, then Cay+(G, S) is the union of |G/H| complete subgraphs isomorphic to K|H|.

If p ≥ 3, then there are |G/H| − 1 bipartite subgraphs of order 2|H| and one component of order

|H|. In particular, if S ≤ G, then C(0) is the complete subgraph isomorphic to K|H| and there are

|G/H| − 1 complete bipartite subgraphs isomorphic to K|H|,|H|.

Recall that the girth of a graph is deﬁned as the length of a shortest cycle in the graph. If the graph does not contain any cycles, then its girth is deﬁned to be ∞. ON INTEGRAL CAYLEY SUM GRAPHS 589

Lemma 2.8 — Let G1 and G2 be two ﬁnite abelian groups, S1 ⊆ G1 and S2 ⊆ G2. Then,

+ + + girth(Cay (G1 ⊕ G2,S)) ≤ min{girth(Cay (G1,S1)), girth(Cay (G2,S2))}, where S = {(s1, 0) | s1 ∈ S1} ∪ {(0, s2) | s2 ∈ S2}.

+ + PROOF : Without loss of generality, we assume that k = girth(Cay (G1,S1)) ≤ girth(Cay + (G2,S2)) and g1 ∼ g2 ∼ · · · ∼ gk ∼ g1 is a shortest cycle in Cay (G1,S1). So (g1, 0) ∼ (g2, 0) ∼ + + · · · ∼ (gk, 0) ∼ (g1, 0) is a cycle in Cay (G1 ⊕ G2,S) of length k and thus girth(Cay (G1 ⊕

G2,S)) ≤ k, as desired. 2

In the following example we determine the girth of a Cayley sum graph with respect to the set of all elements of prime order.

Example 2.9 : Let G = Cay+(G, S), where G is a ﬁnite abelian group and S is the set of all elements of order p, where p varies over all prime divisors of |G|. By the fundamental theorem of

∼ α α α α α ﬁnite abelian groups we have G = Zp 11 ⊕ · · · ⊕ Z 1r1 ⊕ Zp 21 ⊕ · · · ⊕ Z 2r2 ⊕ · · · ⊕ Zp mrm , for 1 p1 2 p2 m some m ∈ N; ri and αiri , i = 1, . . . , m, are positive integers, and pi’s are distinct prime divisors of |G|. Then, Xm r1 r2 rm rj |S| = (p1 − 1) + (p2 − 1) + ··· + (pm − 1) = pj − m j=1 and hSi =∼ Z ⊕ · · · ⊕ Z ⊕ · · · ⊕ Z ⊕ · · · ⊕ Z . (1) | p1 {z p}1 | pm {z pm} r1−times rm−times

We investigate some properties of G. First we claim that G is connected if and only if G = hSi. If G is connected, then by Lemma 2.2, G = hSi, as required. To prove the converse, suppose that G = hSi. By Lemma 2.2, it is enough to show that |G : hS − Si| ≤ 2. According to the parity of |G| we consider two cases.

Case 1 : |G| is odd. Then for every s ∈ S we have 2s ∈ S as ord(s) is an odd prime and S contains all elements of prime order. Hence s = 2s − s ∈ S − S and thus S ⊆ hS − Si. Therefore, G = hSi ≤ hS − Si and G = hS − Si.

Case 2 : |G| is even. According to the number of Z2’s in (1), we have two subcases.

Subcase 1 : There are more than one Z2 in (1). If s1 is an element of S of order two, then there exists s2 in S of order 2 such that s1 = (s1 + s2) − s2 ∈ S − S. If s is an element of S of prime odd order, then s = 2s − s ∈ S − S. Therefore, S ⊆ hS − Si which implies that G = hSi ≤ hS − Si and so G = hS − Si. 590 MARZIEH AMOOSHAHI AND BIJAN TAERI

Subcase 2 : There is exactly one Z2 in (1). Then G = Z2 ⊕Zq2 ⊕· · ·⊕Zqm , where qi’s are distinct odd primes. Let s be the unique element of S of order 2. By previous case, S − {s} ⊆ hS − Si and |G| thus 2 = |hS − {s}i| ≤ |hS − Si|, as required. In what follows, using Lemma 2.8, we show that girth(G) ∈ {3, 4, 6, ∞}. Consider the following cases.

Case 1 : 2 and 3 are the only prime divisors of |G|. Then we have the following subcases.

Subcase 1 : G = Z2 or G = Z3. Then, girth(G) = ∞.

Subcase 2 : G is an elementary abelian 2−group such that |G| > 2. Then G is isomorphic to

K|G|. So, girth(G) = 3.

Subcase 3 : G is an elementary abelian 3−group such that |G| > 3. Then 0 ∼ s1 ∼ s2 − s1 ∼ 0 is a cycle of G, for some s1 ∈ S and s2 ∈ G \ hs1i. Therefore, girth(G) = 3.

n−1 Subcase 4 : G = Z2n and n ≥ 2. Then the components of G are 2 − 1 paths of length 1 and two semi-edges. Thus, girth(G) = ∞. ∼ Subcase 5 : G = Z3m and m ≥ 2. Then, by Theorem 2.5, for every x, y ∈ G\hSi, C(x) = C(y), where C(x) is x ∼ 3m−1 − x ∼ 3m−1 + x ∼ −x ∼ 2.3m−1 + x ∼ 2.3m−1 − x ∼ x, a cycle of length 6. In fact G has (3m−1 −1)/2 components isomorphic to the cycle of length 6 and a component + isomorphic to Cay (Z3, {1, 2}). So, girth(G) = 6.

Subcase 6 : G = Z2t ⊕ Z3s , t ≥ 2 and s ≥ 1. Then,

(2t−2, 2.3s−1) ∼ (2t−2, 3s−1) ∼ (3.2t−2, 0) ∼ (2t−2, 2.3s−1) is a cycle of length 3. Therefore, girth(G) = 3.

Subcase 7 : G = Z2 ⊕ Z3s and s ≥ 1. Then,

(0, 0) ∼ (1, 0) ∼ (1, 3s−1) ∼ (0, 2.3s−1) ∼ (0, 0) is a cycle of length 4. Therefore, girth(G) ≤ 4. In particular, for s = 1, girth(G) = 4.

n m Subcase 8 : G = Z2 ⊕ Z3 ⊕ Z2t ⊕ Z3s , n, m, t, s ∈ N ∪ {0} and G is not isomorphic to the groups mentioned in the previous subcases. Then, by Lemma 2.8 and previous subcases, we conclude that girth(G) ≤ 4.

Case 2 : |G| has a prime divisor p ≥ 5. Then 0 ∼ s ∼ 2s ∼ 0 is a cycle in G, where s ∈ S is an element of order p. So, girth(G) = 3. ON INTEGRAL CAYLEY SUM GRAPHS 591

3. WHICH GROUPSARE CAYLEY SUM INTEGRAL?

Let f : G −→ C be a complex valued function on G. A subset S of G is called f−integral if X f(S) := f(s) s∈S is an integer.

Let G be a ﬁnite abelian group of order n. A character χ of G is a homomorphism from G into the multiplicative group of complex numbers, i.e. χ : G → C∗ such that for every g, h ∈ G and r, s ∈ Z we have χ(rg + sh) = (χ(g))r(χ(h))s.

So for every g ∈ G, (χ(g))n = χ(ng) = χ(0) = 1 implies that χ(g) is an n−th root of unity.

Let A1,A2,...,An be subsets of a ﬁnite non-empty set M. The Boolean algebra generated by

A1,A2,...,An in M is the smallest system of subsets of M that contains A1,A2,...,An and is invariant under the set operations union, intersection and forming the complement that is denoted by

B(A1,A2,...,An; M). Also, the Boolean algebra generated by the subgroups of G is denoted by B(G). For more details, see [17]. We need the following lemma which states that every element of B(G) is χ-integral for all characters χ of G.

Lemma 3.1 [14] — For an arbitrary character χ of G, every set S ∈ B(G) is χ−integral.

Also we need the following lemma on the eigenvalues of Cayley sum graphs.

Lemma 3.2 [10] — Let S be a subset of G. Let R = {χa | a + a = 0} be the set of real- valued characters of G and C be a set containing exactly one character from each conjugate pair + {χa, χ−a}, where a ∈ G and a + a 6= 0. The multi-set of eigenvalues of Cay (G, S) is equal to {χ(S) | χ ∈ R} ∪ {±|χ(S)| | χ ∈ C}.

Recall that a graph is called integral whenever all eigenvalues of the adjacency matrix are integers. Evidently Cay+(G, G) is an integral graph with spectrum {0|G|−1, |G|} (0 with multiplicity |G| − 1 and |G| with multiplicity 1). In the following theorem, we ﬁnd subsets of G that leads to non-trivial integral Cayley sum graphs of G.

Theorem 3.3 — If S ∈ {A + g | A ∈ B(G), g ∈ G}, then Cay+(G, S) is integral.

PROOF : According to Lemma 3.2, all eigenvalues of Cay+(G, S) are χ(S), where χ ∈ R and ±|χ(S)|, where χ ∈ C. Suppose that S = A + g, where A ∈ B(G) and g ∈ G. Then, χ(S) = χ(A)χ(g). If χ is a real-valued character, then χ(g) = ±1 and thus χ(S) is an integer, by 592 MARZIEH AMOOSHAHI AND BIJAN TAERI

Lemma 3.1. If χ is a non real-valued character, then |χ(S)| = |χ(A)| |χ(g)| = |χ(A)|. Now since χ(g) is an n−th root of unity, |χ(g)| = 1 and thus |χ(S)| is an integer, by Lemma 3.1.

The following example shows that there exist many non-trivial and interesting integral Cayley sum graphs on ﬁnite abelian groups.

Example 3.4 : Let G be a ﬁnite abelian group G and S be the subset of all elements of prime order. Then S ∈ B(G). Since

[ ³ ´ ³ [ ´ S = hxi ∩ {0}c = hxi ∩ {0}c, x∈S x∈S by Theorem 3.3, G = Cay+(G, S) is integral.

Deﬁnition 3.5 — A ﬁnite abelian group G is called k−integral, 1 ≤ k ≤ |G|, if and only if for every subset S of G of size k, Cay+(G, S) is integral.

It is clear that, a ﬁnite abelian group G is Cayley sum integral if and only if G is k−integral, for every 1 ≤ k ≤ |G|.

Lemma 3.6 — If Zn is a Cayley sum integral group, then n ≤ 3.

∗ PROOF : Consider the character χ : Zn −→ C given by 1 7→ exp(2πi/n) and put S = {0, 1}. π π Then χ(S) = 1 + exp(2πi/n) and so |χ(S)| = 2 cos n must be an integer. Thus 2 cos n ∈ {0, 1, 2} which implies that n ∈ {1, 2, 3}.

Theorem 3.7 — Every ﬁnite elementary abelian 2-group is Cayley sum integral.

PROOF : Let χ be a character of a ﬁnite abelian group G and g ∈ G be an element of order 2. Then (χ(g))2 = χ(2g) = χ(0) = 1, which implies that χ(g) = ±1. Now, suppose that G is a ﬁnite elementary abelian 2-group. Thus, for every subset S and for every character χ of G, χ(S) is an integer. So the result follows.

Lemma 3.8 — Let G1 and G2 be ﬁnite abelian groups. If G1 ⊕ G2 is Cayley sum integral, then

G1 is Cayley sum integral.

PROOF : Suppose that G1 is not Cayley sum integral, then there exist S1 ⊆ G1 and a character χ ∗ of G1 such that |χ(S1)| ∈/ Z. Using this character, we conclude that the mapping ψ : G1⊕G2 −→ C , given by (g1, g2) 7→ χ(g1), is a character of G1 ⊕ G2. Consider S = {(s1, 0) | s1 ∈ S1} as a subset of G1 ⊕ G2. Then ψ(S) = χ(S1) ∈/ Z. Therefore, G1 ⊕ G2 is not a Cayley sum integral group, that is a contradiction. ON INTEGRAL CAYLEY SUM GRAPHS 593

Theorem 3.9 — A ﬁnite abelian group G is Cayley sum integral if and only if G is an elementary ∼ abelian 2-group or G = Z3.

PROOF : Suppose that G is a Cayley sum integral group and let Zt be an indecomposable factor of the decomposition of G into cyclic factors. By Lemma 3.6, t ≤ 3. To complete the proof, we must ∗ show that Z3 ⊕ Z3 is not Cayley sum integral. Consider the character χ : Z3 ⊕ Z3 −→ C , given by √ √ 3 3i (r, s) 7→ exp(2πri/3). Let S = {(1, 0), (1, 2), (2, 1)}. Then |χ(S)| = | − 2 − 2 | = 3 ∈/ Z and n thus, by Lemma 3.8, for every n ≥ 2, Z3 is not Cayley sum integral.

The converse is a straightforward result of Theorem 3.3 and Theorem 3.7. 2 ∼ n ∼ n Lemma 3.10 — A ﬁnite abelian group G is 2−integral if and only if G = Z2 or G = Z3 , for all n ≥ 0.

PROOF : (=⇒) Let Zn1 ⊕ · · · ⊕ Znt be the decomposition of G into indecomposable cyclic factors, where ni, t ∈ N, i = 1, 2, . . . , t. Since G is 2−integral, Zn1 is 2−integral. Let S = {0, 1} ∗ and χ : Zn1 −→ C , given by 1 7→ w, where w = exp(2πi/n1), is a character of Zn1 which extends to a character of G. We have |χ(S)| = |1 + w| = 2 cos( π ). Now Cay+(Z ,S) is integral if and n1 n1 only if cos( π ) ∈ {± 1 , 0, ±1} if and only if n ∈ {1, 2, 3}. But by the proof of Lemma 3.6, Z ⊕ Z n1 2 1 2 3 ∼ n ∼ n is not 2−integral. Thus, by Lemma 3.8, G = Z2 or G = Z3 , where n ≥ 0. n ∼ n (⇐=) By Theorem 3.9, the groups Z and Z3 are 2−integral. Therefore, suppose that G = Z , n ≥ 0 2 √ √ 3 −1 3 −1 3 and S ⊆ G such that |S| = 2. Let {1, λ = 2 + 2 i, µ = 2 − 2 i} be the set of 3−th roots of unity. So each eigenvalue of Cay+(G, S) comes from the set Γ = {2, 1 + λ, 1 + µ, 2λ, λ + µ, 2µ}. Now, since for each γ ∈ Γ, |γ| is an integer, the result follows. 2 ∼ n m ∼ n m Lemma 3.11 — If G is 3−integral, then G = Z2 ⊕ Z3 or G = Z2 ⊕ Z4 , where n ≥ 0 and m ∈ {0, 1}.

PROOF : Let Zn1 ⊕ · · · ⊕ Znt be the decomposition of G into indecomposable cyclic factors. ∗ Since G is 3−integral, Zn1 is 3−integral. Let S = {0, 1, n1 − 1} and χ : Zn1 −→ C , given by

1 7→ w, where w = exp(2πi/n1) is a character of Zn1 which extends to a character of G. We have µ ¶ 2π χ(S) = 1 + w + wn1−1 = 1 + w + w = 1 + 2Re(w) = 1 + 2 cos . n1

Now Cay+(Z ,S) is integral if and only if cos( 2π ) ∈ {± 1 , 0, ±1} if and only if n ∈ {1, 2, 3, 4}. n1 n1 2 1 On the other hand, by the proof of Theorem 3.9, Z3 ⊕ Z3 is not 3−integral. Also, by considering

{(1, 0), (2, 1), (0, 1)} as a subset of both Z4 ⊕ Z4 and Z3 ⊕ Z4, one can easily see that the groups ∼ n m ∼ n m Z4 ⊕ Z4 and Z3 ⊕ Z4 are not 3−integral. Then, by Lemma 3.8, G = Z2 ⊕ Z3 or G = Z2 ⊕ Z4 such that n ≥ 0 and m ∈ {0, 1}. 2 594 MARZIEH AMOOSHAHI AND BIJAN TAERI

Figure 1: The components of integral 2-regular Cayley sum graphs.

+ + Figure 2: Cay (Z2m, {1, 2m − 1}) and Cay (Zn, {0, 1}), from left to right.

Recall from Lemma 2.2 that Cay+(G, S) is connected if and only if hSi = G and |G : hS − Si| ≤ 2. In the rest of the paper we use this fact frequently without explicit reference.

+ Lemma 3.12 — There exists a subset S of Zn of size two such that Cay (Zn,S) is connected and integral if and only if n ∈ {2, 3, 4, 6}.

+ + PROOF : Suppose Cay (Zn,S) is connected. Then since |S| = 2, Cay (Zn,S) is a cy- + cle or a path with two semi-edges. By Propositions 2.3 and 2.4, Cay (Zn,S) is isomorphic to + + Cay (Z2m, {1, 2m − 1}) or Cay (Zn, {0, 1}), where m ∈ N, see Figure 1.

By [6], the only integral cycles with even vertices are ones with 4 and 6 vertices. On the other + hand, a set of eigenvalues of Cay (Zn, {0, 1}) is

n {±|χ(0) + χ(1)| | χ is a non − real character of Zn} = {|1 + w| | w = 1} kπ = {|1 + exp(2kπi/n)| | 1 ≤ k ≤ n} = {2 cos | 1 ≤ k ≤ n}. n

kπ kπ 1 Therefore |1 + w| = 2 cos n is an integer if and only if cos n ∈ {0, ± 2 , ±1} if and only if n ∈ + {2, 3}. Thus, a connected Cayley sum graph Cay (Zn,S) is integral if and only if it is isomorphic + + + + to Cay (Z4, {1, 3}), Cay (Z6, {1, 5}), Cay (Z2, {0, 1}) or Cay (Z3, {0, 1}), see Figure 2. 2

+ Corollary 3.13 — For every subset S of Zn of size two, Cay (Zn,S) is connected and integral if and only if n ∈ {2, 3}. ON INTEGRAL CAYLEY SUM GRAPHS 595

+ PROOF : By Lemma 3.12, there exists a subset S of Zn of size two such that Cay (Zn,S) is connected and integral if and only if n ∈ {2, 3, 4, 6}. By Theorem 3.9, Z2 and Z3 are integral, so it is enough to consider n = 4 and n = 6. Put S1 = {0, 1} ⊆ Z4 and S2 = {0, 1} ⊆ Z6. It is easy to see √ √ + + that 2 and 3 are eigenvalues of Cay (Z4,S1) and Cay (Z6,S2), respectively (we may use the + second GAP code given in the appendix). Therefore, Cay (Zn,S) is connected and integral if and only if n ∈ {2, 3}. 2

We need the classiﬁcation of simple connected cubic integral graphs given by Bussemaker and Cvetkovic´ [7].

Lemma 3.14 [7] — There are exactly 13 connected cubic integral graphs, say Γi, 1 ≤ i ≤ 13. The spectrum of these graphs are: 4 5 4 Γ1 = K4, Γ2 with spectrum {3, 0 , −3}, Γ3 with spectrum {3, 1 , −2 }, Γ4 with spectrum 3 3 2 2 9 10 9 {3, 1 , −1 , −3}, Γ5 with spectrum {3, 1, 0 , −2 }, Γ6 with spectrum {3, 2 , 0 , −2 , −3}, Γ7 with 3 2 3 3 2 3 3 4 spectrum {3, 2, 1 , −1 , −2 }, Γ8 with spectrum {3, 2 , 0 , −1 , −2 }, Γ9 with spectrum {3, 2 , 5 5 4 4 5 5 4 2 2 1 , −1 , −2 , −3}, Γ10 with spectrum {3, 2 , 1 , −1 , −2 , −3}, Γ11 with spectrum {3, 2, 1 , 0 , 2 2 4 2 6 −1 , −2, −3}, Γ12 with spectrum {3, 2 , 1, 0 , −1, −2 , −3} and the graph Γ13 with spectrum {3, 2 , 13, 04, −13, −26, −3}.

+ Lemma 3.15 — There exists a square free subset S of Zn of size three such that Cay (Zn,S) is connected and integral if and only if n ∈ {6, 8, 12}.

PROOF : Since S is a square free subset of Zn, Lemma 3.14 implies that n ∈ {4, 6, 8, 10, 12, 20, 24, 30}. Using the ﬁrst GAP code given in the appendix, we ﬁnd that there exists a square free subset + S of size 3 in Zn such that Cay (Zn,S) is connected and integral if and only if n ∈ {6, 8, 12}. This completes the proof. 2

+ Corollary 3.16 — Cay (Zn,S) is connected and integral, for all subset S of Zn of size three, if and only if n ∈ {6, 8}.

PROOF : First note that the only square free subset of Z6 of size 3 is {1, 3, 5}. It is easy to + see that Cay (Z6, {1, 3, 5}) is connected and integral. Also all square free subsets of Z8 of size 3 are S1 = {1, 3, 5}, S2 = {1, 3, 7}, S3 = {1, 5, 7} and S4 = {3, 5, 7}. For each i ∈ {1, 2, 3, 4}, + Cay (Z8,Si) is connected and integral (we may use the ﬁrst GAP code given in the appendix). + Now suppose that Cay (Zn,S) is connected and integral for all subsets of size three. By Lemma + 3.15, n ∈ {6, 8, 12}. We note that Cay (Z12,S) is not integral for some subset S of size three. In √ + fact it is easy to see that 3 is an eigenvalue of Cay (Z12, {1, 3, 7}) (we may use the second GAP code given in the appendix). Hence n ∈ {6, 8}. 2 596 MARZIEH AMOOSHAHI AND BIJAN TAERI

In order to prove the main result on cubic integral Cayley sum graph, we need the following lemma, which is proved by Cheyne et al. [8].

Lemma 3.17 [8] — Let S be a square free subset of a ﬁnite abelian group G such that Cay+(G, S) is connected. Then, the following statements are equivalent:

(i) Cay+(G, S) is a bipartite graph;

(ii) |G : hS − Si| = 2;

(iii) S ∩ hS − Si = ∅.

Furthermore, if Cay+(G, S) is bipartite, then hS − Si and G \ hS − Si are the bipartition sets.

From the following theorem we see that if G is a ﬁnite abelian group, then for some square free subset S of size 3, Cay+(G, S) is connected and integral if and only if G is isomorphic to one of the following groups:

2 3 Z2, Z6, Z8, Z2 ⊕ Z4, Z2, Z12, Z2 ⊕ Z6, Z2 ⊕ Z12, Z4 ⊕ Z6, Z2 ⊕ Z2 ⊕ Z6.

Theorem 3.18 — There are exactly six simple connected cubic integral Cayley sum graphs.

PROOF : Let Cay+(G, S) be a simple connected cubic integral graph. Then by Lemma 3.14, + + Cay (G, S) is of type Γi, 1 ≤ i ≤ 13. We show that Cay (G, S) is of type Γ1, Γ2, Γ4, Γ8, Γ12,

Γ13. Since the number of vertices of Γi belongs to {4, 6, 8, 10, 12, 20, 24, 30}, 1 ≤ i ≤ 13, we have the following cases: ∼ Case 1 : Let |G| = 4. Since Z4 has no square free subset of size 3, we may assume that G =

Z2 ⊕ Z2. Now the only square free subset of Z2 ⊕ Z2 of size 3 is S = {(1, 0), (1, 1), (0, 1)}; and we + ∼ have Cay (Z2 ⊕ Z2,S) = K4 = Γ1.

Case 2 : Let |G| = 6. By Lemma 3.16 and ﬁrst GAP code given in the appendix, we have + Cay (Z6, {1, 3, 5}) = Γ2. Now since Z6 is the only abelian group of order 6, Γ5 is not a Cayley sum graph.

Case 3 : Let |G| = 8. Then, G is isomorphic to Z8, Z2 ⊕ Z4 or Z2 ⊕ Z2 ⊕ Z2. By Corollary + 3.16, for every square free subset S of Z8 of size 3, Cay (Z8,S) is connected and integral. Then + Cay (Z8,S) = Γ4. By Theorem 3.9, Z2 ⊕ Z2 ⊕ Z2 is a Cayley sum integral group and so for every + ∼ square free subset S of Z2 ⊕ Z2 ⊕ Z2, Cay (Z2 ⊕ Z2 ⊕ Z2,S) = Γ4. Finally, let G = Z2 ⊕ Z4. Then + S = {(0, 1), (1, 2), (1, 0)} is a subset of Z2 ⊕ Z4 and Cay (Z2 ⊕ Z4,S) is connected and integral + and so, Cay (Z2 ⊕ Z4,S) = Γ4. ON INTEGRAL CAYLEY SUM GRAPHS 597

Case 4 : Let |G| = 10. By Lemma 3.15, Z10 has no square free subset S of size 3 such that + Cay (Z10,S) is connected and integral. So the graphs Γ3 (Petersen graph), Γ7 and Γ11 are not Cayley sum graphs.

Case 5 : Let |G| = 12. By Lemma 3.15, there exists a square free subset S of Z12 of size 3 + + such that Cay (Z12,S) is connected and integral. By Lemma 3.14, Cay (Z12,S) = Γ12 or Γ8. + Since for each square free subset of Z12 we have 2 ∈ S − S, Lemma 3.17 implies that Cay (Z12,S) is bipartite. But Bussemaker and Cvetkovic´ [7] showed that Γ8 is not bipartite and Γ12 is bipartite. + ∼ Hence Cay (Z12,S) = Γ12. Now suppose that G = Z2 ⊕ Z6. By second GAP code given in the + + appendix, Cay (Z2 ⊕Z6, {(1, 3), (1, 2), (0, 1)}) = Γ8 and Cay (Z2 ⊕Z6, {(1, 0), (0, 5), (0, 1)}) =

Γ12.

+ Case 6 : Let |G| = 20. By Lemma 3.15, Z20 has no square free subset S such that Cay (Z20,S) ∼ is connected and integral. Now, let G = Z2 ⊕ Z10. By making a few modification in the first GAP + code given in the appendix, we find that Z10 has no subset S of size 3 such that Cay (Z10,S) is + integral. It remains to prove that for every subset S1 of size 2 of Z10, such that Cay (Z10,S1) is + integral, Cay (Z2 ⊕ Z10,S) is not a simple connected integral graph, where S ⊆ Z2 × S1 and |S| = 3. Again,by making a few modification in the first GAP code given in the appendix, we find + that all subsets S1 of size 2 of Z10 such that Cay (Z10,S1) is integral are contained in n o T = {0, 5}, {1, 6}, {2, 7}, {3, 8}, {4, 9} .

Consider the set O = {S | S ⊆ Z2 × S1,S1 ∈ T, |S| = 3 and S is square free}. It is easy to see + that, for every S ∈ O, Cay (Z2 × Z10,S) is not a connected graph. So, the graphs Γ9 and Γ10 are not Cayley sum graphs.

Case 7 : Let |G| = 24. By Lemma 3.15, Z24 has no square free subset S of size 3 such that Cay+(G, S) is connected and integral. Using the second GAP code given in the appendix, the Cay- + + ley sum graphs Cay (Z2 ⊕ Z12, {(0, 11), (1, 3), (1, 1)}), Cay (Z4 ⊕ Z6, {(1, 0), (1, 1), (3, 2)}) and + Cay (Z2 ⊕ Z2 ⊕ Z6, {(1, 1, 5), (1, 1, 4), (1, 0, 2)}) are connected and integral isomorphic to Γ13.

Case 8 : Let |G| = 30. By Lemma 3.15, Z30 has no square free subset of size 3 such that + Cay (Z30,S) is connected and integral. So, the graph Γ6 is not a Cayley sum graph. 2

Corollary 3.19 — For every square free subset S of G of size 3, Cay+(G, S) is connected and integral if and only if G is isomorphic to one of the following groups

2 3 Z2, Z2, Z6, Z8. 598 MARZIEH AMOOSHAHI AND BIJAN TAERI

PROOF : By Theorem 3.18, it is enough to consider the following groups

Z2 ⊕ Z4, Z2 ⊕ Z6, Z2 ⊕ Z12, Z4 ⊕ Z6, Z2 ⊕ Z2 ⊕ Z6.

√ Now, by using the second GAP code given in the appendix, we can see that 5 is an eigenvalue of the Cayley sum graphs

+ + Cay (Z2 ⊕ Z4, {(0, 1), (1, 3), (1, 0)}) and Cay (Z4 ⊕ Z6, {(1, 0), (1, 1), (3, 2)}), √ 3 is an eigenvalue of

+ + Cay (Z2 ⊕ Z6, {(1, 1), (0, 1), (1, 5)}) and Cay (Z2 ⊕ Z12, {(1, 0), (0, 1), (1, 6)}), and √ + 7 is an eigenvalue of Cay (Z2 ⊕ Z2 ⊕ Z6, {(1, 0, 0), (0, 1, 0), (0, 0, 1)}). Thus, by Theorem 3.9 and Corollary 3.16, the result follows. 2

ACKNOWLEDGEMENT

The authors gratefully thank to referee for the helpful comments and recommendations leading us to improve the readability and quality of the paper.

REFERENCES

1. A. Abdollahi and E. Vatandoost, Which Cayley graphs are integral?, Electronic Journal of Combina- torics, 16(1) (2009), #R122. 2. A. Ahmadi, R. Belk, C. Tamon and C. Wendler, On mixing of continuous-time quantum walks on some circulant graphs, Quantum Inform. and Comp., 3 (2003), 611-618. 3. N. Alon, Large sets in ﬁnite ﬁelds are sumsets, J. Number Theory, 126(1) (2007), 110-118. 4. M. Amooshahi and B. Taeri, On the domination and total domination numbers of Cayley sum graphs

over Zn, Kragujevac J. Math., 38(2) (2014), 315-320. 5. M. Amooshahi and B. Taeri, On Cayley sum graphs of non-abelian groups, Graphs and Combinatorics, 32(1) (2016), 17-29. 6. K. Balin´ ska, D. Cvetkovic,´ Z. Radosavljevic,´ S. Simic´ and D. Stevanovic,´ A survey on integral graphs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 13 (2002), 42-65. 7. F. C. Bussemaker and D. Cvetkovic,´ There are exactly 13 connected, cubic, integral graphs, Univ. Beograd, Publ. Elektrotehn. Fak. Ser. Mat. Fiz., 544 (1976), 43-48. 8. B. Cheyne, V. Gupta and C. Wheeler, Hamilton cycles in addition graphs, Rose-Hulman Undergraduate Math. Journal, 1(4) (2003). ON INTEGRAL CAYLEY SUM GRAPHS 599

9. F. R. K. Chung, Diameters and eigenvalues, J. Amer. Math. Soc., 2(2) (1989), 187-196. 10. M. Devos, L. Goddyn, B. Mohar and R. Samal,´ Cayley sum graphs and eigenvalues of (3,6)-fullerenes, J. Combin. Theory Ser. B, 99 (2009), 358-396. 11. B. Green, Counting sets with small sumset, and the clique number of random Cayley graphs, Combina- torica, 25 (2005), 307-326. 12. D. Grynkiewicz, V. F. Lev and O. Serra, Connectivity of addition Cayley graphs, J. Comin. Theory Ser. B, 99 (2009), 202-217. 13. F. Harary and A. Schwenk, Which graphs have integral spectra?, in: R. Bari, F. Harary (Eds.), Graphs and Combinatorics, Springer, Berlin (1974). 14. W. Klotz and T. Sander, On integral Cayley graphs over abelian groups, Notes Number Theory Discrete Math., 17(3) (2011), 49-59. 15. V. F. Lev, Sums and differences along Hamiltonian cycles, Discrete Math., 310 (2010), 575-584. 16. N. Saxena, S. Severini and I. E. Shparlinski, Parameters of integral circulant graphs and periodic quantum dynamics, Int. J. Quant. Inf., 5 (2007), 417-430. 17. R. Sikorski, Boolean Algebras, Springer-Verlag, Berlin, Heidelberg, New York (1969). 18. D. Sinha, P. Garg and A. Singh, Some properties of unitary addition Cayley graphs, Notes Number Theory Discrete Math., 17(3) (2011), 49-59. 19. The GAP Group, GAP-Groups, Algorithms, and Programming, Version 4.4.4, 2004, http://gap-system.org.

APPENDIX

The following code, written in the computer algebra system GAP [19], by using Lemma 2.2 and + Lemma 3.2, determines all subsets S of Zn of size 3 such that Cay (Zn,S) is connected and integral, + and returns all such subsets S with the absolute value of eigenvalues of Cay (Zn,S).

Main:=function(n) local i,j,k,r,t,a,c,w,A,B,G,T,D; G:=CyclicGroup(IsPermGroup,n); a:=MinimalGeneratingSet(CyclicGroup(IsPermGroup,n))[1]; T:=Elements(List(G,x->xˆ2)); B:=Set([]); for i in [1..n-1] do for j in [i+1..n-1] do for k in [j+1..n-1] do w:=0; A:=[]; D:=[]; for r in [0..n-1] do c:=E(n)ˆ(i*r)+E(n)ˆ(j*r)+E(n)ˆ(k*r); t:=ImaginaryPart(c)ˆ2+RealPart(c)ˆ2; 600 MARZIEH AMOOSHAHI AND BIJAN TAERI

if IsInt(t) and IsInt(Sqrt(t)) then w:=w+1; fi; od; if w=n and Intersection([aˆi,aˆj,aˆk],T)=[] and Group(aˆi,aˆj,aˆk)=G and Size(Group(aˆ(i-j),aˆ(i-k),aˆ(j-k)))>=Size(G)/2 then Add(D,i); Add(D,j); Add(D,k); Add(A,D); for r in [0..n-1] do c:=E(n)ˆ(i*r)+E(n)ˆ(j*r)+E(n)ˆ(k*r); t:=ImaginaryPart(c)ˆ2+RealPart(c)ˆ2; if IsInt(t) then t:=Sqrt(t); fi; Add(A,t); od; Add(B,A); fi; od; od; od; return B; end;

The following table is the result of above code, which is used in Lemma 3.15 and Corollary 3.16.

+ n S Spec(Cay (Zn,S)) 6 { 1, 3, 5 } {−3, 04, 3} { 1, 3, 5 } { 1, 3, 7 } 8 {−3, −13, 13, 3} { 1, 5, 7 } { 3, 5, 7 } { 1, 3, 5 } { 1, 3, 7 } { 1, 9, 11} 12 {−3, −22, −1, 04, 1, 22, 3} { 3, 5, 11} { 5, 7, 9 } { 7, 9, 11} ON INTEGRAL CAYLEY SUM GRAPHS 601

+ The following algorithm, by using Lemma 3.2, gives the absolute value of eigenvalues of Cay (Zn ⊕

Zm ⊕ Zr,S), where S is a subset of Zn ⊕ Zm ⊕ Zr.

Test:=function(n,m,r,S) local i,j,k,t,c,A; A:=[]; for i in [0..n-1] do for j in [0..m-1] do for t in [0..r-1] do c:=0; for k in [1..Size(S)] do c:=c+E(n)ˆ(i*S[k][1])*E(m)ˆ(j*S[k][2])*E(r)ˆ(t*S[k][3]); od; Add(A,Sqrt(ImaginaryPart(c)ˆ2+RealPart(c)ˆ2)); od; od; od; return A; end;