Groups all of whose undirected Cayley graphs are integral

Let GG be a finite group, S⊆G∖{1}S⊆G∖{1} be a set such that if a∈Sa∈S, then a−1∈Sa−1∈S, where 11 denotes the identity element of GG. The undirected Cayley graph Cay(G,S)Cay(G,S) of GG over the set SS is the graph whose vertex set is GG and two vertices aa and bb are adjacent whenever ab−1∈Sab−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 GG Cayley integral whenever all undirected Cayley graphs over GG are integral. Finite abelian Cayley integral groups are classified by Klotz and Sander as finite abelian groups of exponent dividing 44 or 66. 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 S3S3 of degree 33, C3⋊C4C3⋊C4 and Q8×C2n for some integer n≥0n≥0, where Q8Q8 is the quaternion group of order 88.