Some Algebraic Properties of Bi-Cayley Graphs

For a finite group G and a subset S(possibly,it contains the identity element) of G,the Bi-Cayley graph X=BC(G,S) of G with respect to S is defined as the bipartite graph with vertex set G×{0,1} and edge set {{(g,0),(sg,1)}:g∈G,s∈S}. In this paper,we investigate the relation between the eigenvaiues of Cayley graph D(G,S) and Bi-Cayley graph BC(G,S) for a finite abelian group.As a consequence, we determine the eigenvalues of Bi-Cayley graphs of cyclic groups.In addition,some asymptotic enumeration theorems are presented.