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\subseteq G‎, ‎which is denoted by BCay(G,S)‎, ‎is the graph with‎ ‎vertex set G\times\{۱,۲\} and edge set \{\{(x,۱)‎, ‎(sx,۲)\}\mid x\in G‎, ‎\ s\in S\}‎. ‎A‎ ‎finite group G is called a \textit{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 \Bbb Z_۲^k‎, ‎for some k‎, ‎\Bbb Z_۳ or S_۳‎.

Keywords:

bi-Cayley graph , Integer Eigenvalues , Representations of finite groups