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 subset of G, which is denoted by BCay(G, S), is the graph with vertex set G x {1, 2} and edge set {{(x,1), (sx, 2)} x is an element of G, s is an element of 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(2)(k) for some k, Z(3) or S-3.