Isomorphisms of bi-Cayley graphs on generalized quaternion groups

TL;DR

This work analyzes the isomorphism problem for bi-Cayley graphs on generalized quaternion groups $\mathrm{Q}_{4n}$, establishing that for $m\in\{2,3\}$, $\mathrm{Q}_{4n}$ is an $m$-BCI-group if and only if $n$ is odd or $n=2$, equivalently an $m$-DCI- and $m$-CI-group. A spectral method based on the irreducible representations of $\mathrm{Q}_{4n}$ distinguishes non-isomorphic BCay graphs, and the authors prove a key non-isomorphism (between $\mathrm{BCay}(\mathrm{Q}_{4n},\{1,a,b\})$ and $\mathrm{BCay}(\mathrm{Q}_{4n},\{1,a^2,b\})$) to motivate the approach. The main result is established via a combination of representation-theoretic spectrum calculations and automorphism-extension arguments, with a detailed case analysis that shows 2- and 3-BCI-properties coincide precisely when $n$ is odd (or $n=2$). Consequently, the paper completes the classification for $m$-BCI-properties of $\mathrm{Q}_{4n}$ in the stated range and reinforces the link between BCI, DCI, and CI properties for this family of nonabelian 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 $T=gS^α$ for some $g\in G$ and $α\in \mathrm{Aut}(G)$. 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 $n=2$.

Theorems & Definitions (24)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• Lemma 2.4
• Remark 2.5
• Proposition 3.1
• Lemma 3.2