A classification of finite groups with integral bi-Cayley graphs

Abstract

The bi-Cayley graph of a finite group G with respect to a subset S⊆G ‎, ‎which is denoted by \BCay(G,S) ‎, ‎is the graph with‎ ‎vertex set G×{1,2} and edge set {{(x,1)‎,‎(sx,2)}∣x∈G‎,‎ s∈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 Z^k_2 ‎, ‎for some k, ‎Z_3 or S_3 ‎.

Keywords

• Bi-Cayley graph
• Integer Eigenvalues
• Representations of finite groups