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.