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

Abstract

Let $kgeq 1$ 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 $1leq |S|leq k$ is integral‎. ‎Let $kgeq 3$‎. ‎We prove that a finite group $G$ belongs to $mathcal{I}_k$ if and‎ ‎only if $GcongBbb Z_3$‎, ‎$Bbb Z_2^r$ for some integer $r$‎, ‎or $S_3$‎.