a classification of finite groups with integral bi-cayley graphs

The bi-Cayley graph of a finite group G with respect to a subset S ⊆ G, which is denoted by BCay(G, S), is the graph with vertex set G × {1, 2} and edge set {{(x, 1), (sx, 2)} | x ∈ G, s ∈ S}. A finite group G is called a bi-Cayley integral group if for any subset S of G, BCay(G, S) is a graph with integer eigenvalues. In this paper we prove that a finite group G is a bi-Cayley integral group if and only if G is isomorphic to one of the groups Z^k 2, for some k, Z3 or S3.