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ON INTEGRAL CAYLEY SUM GRAPHS Marzieh Amooshahi and Bijan Taeri

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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 ON INTEGRAL CAYLEY SUM GRAPHS Marzieh Amooshahi and Bijan Taeri 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 2 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 ffg; hg j g; h 2 G; g + h 2 Sg. If S is a multi-set, then Cay+(G; S) contains multiple edges, and if there exists g 2 G with 2g 2 S, then the edge fg; gg 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 jSj. 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 jGj, and the order of an element g 2 G, is denoted by ord(g). For a subset S of G, let S ¡ S = + fs1 ¡ s2 j s1; s2 2 Sg: If g and h are adjacent in Cay (G; S), i.e. g + h 2 S, we write g » h. Also, N(g) = fh 2 G j g + h 2 Sg 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 jG : Hj = 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 jG : hS ¡ Sij · 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 2 S, as t = (t¡s)+s 2 hS¡Si+s, for every t 2 S. Also, for every s; t 2 S, we have hS ¡ Si + s = hS ¡ Si + t, as s ¡ t 2 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 jG : hS ¡ Sij · 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; f0; 1; 2g) or + Cay (Z2n; f1; 2n ¡ 1g), where n ¸ 3. + + PROOF : First note that Cay (Z3; Z3) and Cay (Z4; f0; 1; 2g) 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; f1; 2n ¡ 1g), n ¸ 3, is a simple cycle of length 2n, since Cay (Z2n; f1; 2n ¡ 1g) is connected and for any vertex u 2 Z2n, we have N(u) = f1 ¡ u; 2n ¡ 1 ¡ ug. To prove the converse, suppose that G = Cay+(G; S) is a cycle. We consider two cases; Case 1 : jGj is odd. Then S ⊆ G = 2G. Thus for any s 2 S, there exists g 2 G such that s = g + g 2 S and so fg; gg is a semi-edge. Since G is a cycle we conclude that every vertex is adjacent to a semi-edge. Thus jSj = 3 and 2G ⊆ S. Therefore G = S and also jGj = jSj = 3. Case 2 : jGj = 2n is even. If jSj = 3, then since G is a cycle, every vertex is adjacent to a semi-edge and so 2G ⊆ S. Therefore jGj = 4 or 6, since j2Gj = n. If jGj = 4, then S = f0; 1; 2g or 586 MARZIEH AMOOSHAHI AND BIJAN TAERI + » + f0; 2; 3g. It is easy to see that Cay (Z4; f0; 1; 2g) = Cay (Z4; f0; 2; 3g) is a cycle that every vertex + is adjacent to a semi-edge. If jGj = 6, then S = f0; 2; 4g. Lemma 2.2 implies that Cay (Z6;S) is not connected. So suppose that jSj = 2. Since G is a cycle we conclude that it has no semi-edges. + Now, since Cay (Z2n; f1; 2n ¡ 1g) is a simple cycle and all simple cycles with the same length are isomorphic, the result holds.
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