Groups all of whose undirected Cayley graphs are integral

Let G be a finite group, S subset of G \ {1} be a set such that if a is an element of S, then a(-1) is an element of S, where I denotes the identity element of G. The undirected Cayley graph Cay(G, S) of G over the set S is the graph whose vertex set is G and two vertices a and b are adjacent whenever ab(-1) is an element of S. The adjacency spectrum of a graph is the multiset of all eigenvalues of the adjacency matrix of the graph. A graph is called integral whenever all adjacency spectrum elements are integers. Following Klotz and Sander, we call a group G Cayley integral whenever all undirected Cayley graphs over G are integral. Finite abelian Cayley integral groups are classified by Klotz and Sander as finite abelian groups of exponent dividing 4 or 6. Klotz and Sander have proposed the determination of all non-abelian Cayley integral groups. In this paper we complete the classification of finite Cayley integral groups by proving that finite non-abelian Cayley integral groups are the symmetric group S-3 of degree 3, C-3 x C-4 and Q(8) x C-2(n) for some integer n > 0, where Q(8) is the quaternion group of order 8. (C) 2013 Elsevier Ltd. All rights reserved.