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 ý, ýwhich is denoted by \BCay(G,S) \ý, ýis the graph withý ývertex set G×{1,2 and edge set {{(x,1)ý,ý(sx,2)}∣x∈Gý,ý s∈S} ý. ýAý ýfinite group G is called a \textit{bi-Cayley integral group} if for any subset S ofýýGý,\BCay(G,S) is a graph with integer eigenvaluesý. ýIn this paper we proveý ýthat a finite group G is a bi-Cayley integral group if and only if G is isomorphic toý ýone of the groups Z2ký, ýfor someký, ý Z3 or S3ý.