Groups all of whose undirected Cayley graphs are integral

Abstract

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 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∈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 S3 of degree 3, C3⋊C4 and Q8×C2n for some integer n≥0, where Q8 is the quaternion group of order 8.