On finite groups all of whose bi-Cayley graphs of bounded valency are integral

Abstract

Let $k\geq 1$ be an integer and $\mathcal{I}_k$ be‎ ‎the set of all finite groups $G$ such that every bi-Cayley graph BCay(G,S) of $G$ with respect to‎ ‎subset $S$ of length $1\leq |S|\leq k$ is integral‎. ‎Let $k\geq 3$‎. ‎We prove that a finite group $G$ belongs to $\mathcal{I}_k$ if and‎ ‎only if $G\cong\Bbb Z_3$‎, ‎$\Bbb Z_2^r$ for some integer $r$‎, ‎or $S_3$‎.

Keywords

• ‎bi-cayley graph‎
• ‎integer eigenvalues‎
• ‎irreducible representation