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arXiv · 2210.06777

Stability of graph pairs involving vertex-transitive graphs

Yan-Li Qin·Binzhou Xia·Sanming Zhou
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Abstract

A pair of graphs $(\Gamma,\Sigma)$ is said to be stable if the full automorphism group of $\Gamma\times\Sigma$ is isomorphic to the product of the full automorphism groups of $\Gamma$ and $\Sigma$ and unstable otherwise, where $\Gamma\times\Sigma$ is the direct product of $\Gamma$ and $\Sigma$. In this paper, we reduce the study of the stability of any pair of regular graphs $(\Gamma,\Sigma)$ with coprime valencies and vertex-transitive $\Sigma$ to that of $(\Gamma,K_2)$. Since the latter is well studied in the literature, this enables us to determine the stability of any pair of regular graphs $(\Gamma,\Sigma)$ with coprime valencies in the case when $\Sigma$ is vertex-transitve and the stability of $(\Gamma,K_2)$ is known.

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Yan-Li Qin, Binzhou Xia, Sanming Zhou. 2022-10-13. Stability of graph pairs involving vertex-transitive graphs. https://arxiv.org/abs/2210.06777

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