Isomorphisms of bi-Cayley graphs on generalized quaternion groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916570939260928 |
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| author | Xie, Jin-Hua Zhang, Zhishuo |
| author_facet | Xie, Jin-Hua Zhang, Zhishuo |
| contents | 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 $n=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11918 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Isomorphisms of bi-Cayley graphs on generalized quaternion groups Xie, Jin-Hua Zhang, Zhishuo Combinatorics 05C25, 20B25 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 $n=2$. |
| title | Isomorphisms of bi-Cayley graphs on generalized quaternion groups |
| topic | Combinatorics 05C25, 20B25 |
| url | https://arxiv.org/abs/2409.11918 |