On integral Cayley sum graphs

Abstract

Let S be a subset of a finite abelian group G. The Cayley sum graph Cay+(G, S) of G with respect to S is a graph whose vertex set is G and two vertices g and h are joined by an edge if and only if g + h ∈ S. We call a finite abelian group G a Cayley sum integral group if for every subset S of G, Cay+(G, S) is integral i.e., all eigenvalues of its adjacency matrix are integers. In this paper, we prove that all Cayley sum integral groups are represented by Z3 and Zn2n, n ≥ 1, where Zk is the group of integers modulo k. Also, we classify simple connected cubic integral Cayley sum graphs.