A classification of finite groups with integral bi-Cayley graphs

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The bi-Cayley graph of a finite group $G$ with respect to a subset $Ssubseteq G$‎, ‎which is denoted by $BCay(G,S)$‎, ‎is the graph with‎ ‎vertex set $Gtimes{1,2}$ and edge set ${{(x,1)‎, ‎(sx,2)}mid xin G‎, ‎ sin S}$‎. ‎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 $Bbb Z_2^k$‎, ‎for some $k$‎, ‎$Bbb Z_3$ or $S_3$‎.