Isomorphisms of bi-Cayley graphs on generalized quaternion groups

TL;DR

This paper characterizes when bi-Cayley graphs on generalized quaternion groups are isomorphic, showing that for valency 2 and 3, these groups are BCI, DCI, and CI groups only under specific conditions related to the order parameter n.

Contribution

It provides a complete characterization of generalized quaternion groups as m-BCI, m-DCI, and m-CI groups for m=2,3, based on the parity of n.

Findings

Generalized quaternion groups of order 4n are m-BCI, m-DCI, and m-CI groups if and only if n is odd or n=2.

The equivalence of m-BCI, m-DCI, and m-CI properties is established for these groups.

The characterization applies specifically to bi-Cayley graphs with valency at most 3.

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 $n=2$.