Isomorphisms of bi-Cayley graphs on generalized quaternion groups

Abstract

Let $G$ be a finite group and $S$ be a subset of $G$. The bi-Cayley graph $\mathrm{BCay}(G,S)$ is the graph with vertex set $G\times \{0,1\}$ and edge set $\{\{(x,0),(sx,1)\}\mid x\in G,s\in S\}$. A bi-Cayley graph $\mathrm{BCay}(G,S)$ is called a BCI-graph if for every $T\subseteq G$, the isomorphism $\mathrm{BCay}(G,S)\cong \mathrm{BCay}(G,T)$ implies that $T=gS^\alpha$ for some $g\in G$ and $\alpha\in \mathrm{Aut}(G)$. We say a group $G$ an $m$-BCI-group if every bi-Cayley graphs of $G$ with valency at most $m$ is a BCI-graph. In this paper, we show that for $m\in\{2,3\}$, the generalized quaternion group of order $4n$ with $n\geq 2$ is an $m$-BCI-group if and only if it is an $m$-DCI-group if and only if it is an $m$-CI-group if and only if $n$ is odd or $n=2$.