Stability of Cayley graphs and Schur rings
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913930590289920 |
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| author | Hujdurović, Ademir Kovács, István |
| author_facet | Hujdurović, Ademir Kovács, István |
| contents | A graph $Γ$ is said to be unstable if for the direct product $Γ\times K_2$, $Aut(Γ\times K_2)$ is not isomorphic to $Aut(Γ) \times \mathbb{Z}_2$. In this paper we show that a connected and non-bipartite Cayley graph $Cay(H,S)$ is unstable if and only if the set $S \times \{1\}$ belongs to a Schur ring over the group $H \times \mathbb{Z}_2$ having certain properties. The Schur rings with these properties are characterized if $H$ is an abelian group of odd order or a cyclic group of twice odd order. As an application, a short proof is given for the result of Witte Morris stating that every connected unstable Cayley graph on an abelian group of odd order has twins (Electron.~J.~Combin, 2021). As another application, sufficient and necessary conditions are given for a connected and non-bipartite circulant graph of order $2p^e$ to be unstable, where $p$ is an odd prime and $e \ge 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_01182 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stability of Cayley graphs and Schur rings Hujdurović, Ademir Kovács, István Combinatorics 05C25, 20B25 A graph $Γ$ is said to be unstable if for the direct product $Γ\times K_2$, $Aut(Γ\times K_2)$ is not isomorphic to $Aut(Γ) \times \mathbb{Z}_2$. In this paper we show that a connected and non-bipartite Cayley graph $Cay(H,S)$ is unstable if and only if the set $S \times \{1\}$ belongs to a Schur ring over the group $H \times \mathbb{Z}_2$ having certain properties. The Schur rings with these properties are characterized if $H$ is an abelian group of odd order or a cyclic group of twice odd order. As an application, a short proof is given for the result of Witte Morris stating that every connected unstable Cayley graph on an abelian group of odd order has twins (Electron.~J.~Combin, 2021). As another application, sufficient and necessary conditions are given for a connected and non-bipartite circulant graph of order $2p^e$ to be unstable, where $p$ is an odd prime and $e \ge 1$. |
| title | Stability of Cayley graphs and Schur rings |
| topic | Combinatorics 05C25, 20B25 |
| url | https://arxiv.org/abs/2308.01182 |