Groups all of whose undirected Cayley graphs are integral

Let G be a finite group, S ⊆ G ∖ { 1 } be a set such that if a ∈ S , then a − 1 ∈ S , where 1 denotes the identity element of G . The undirected Cayley graph C a y ( 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 a b − 1 ∈ 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 ⋊ C 4 and Q 8 × C 2 n for some integer n ≥ 0 , where Q 8 is the quaternion group of order 8 .