On groups admitting no integral Cayley graphs besides complete multipartite graphs

Abstract

Let G be a non-trivial finite group, S ⊇ G\{e} be a set such that if a ∈ S, then a-1 ∈ S and e be the identity element of G. Suppose that Cay(G, S) is the Cayley graph with the vertex set G such that two vertices a and b are adjacent whenever ab-1 ∈ S. An arbitrary graph is called integral whenever all eigenvalues of the adjacency matrix are integers. We say that a group G is Cayley integral simple whenever every connected integral Cayley graph on G is isomorphic to a complete multipartite graph. In this paper we prove that if G is a non-simple group, then G is Cayley integral simple if and only if G ≅ ℤp2 for some prime number p or G ≅ ℤ2×ℤ2. Moreover, we show that there exist finite non-abelian simple groups which are not Cayley integral simple.