A classification of finite groups with integral bi-Cayley graphs

Document Type: Research Paper

Authors

1 Departmant of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran

2 Department of Mathematics, Isfahan University of Technology, Isfahan, Iran

Abstract

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 $G\times\{1,2\}$ and edge set $\{\{(x,1)‎, ‎(sx,2)\}\mid x\in G‎, ‎\ s\in S\}$‎. ‎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 $\Bbb Z_2^k$‎, ‎for some $k$‎, ‎$\Bbb Z_3$ or $S_3$‎.

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