On the order of vertex-stabilisers in vertex-transitive graphs with local group $C_p\times C_p$ or $C_p\wr C_2$

Abstract

Let $p$ be a prime and let $L$ be either the intransitive permutation group $C_p\times C_p$ of degree $2p$ or the transitive permutation group $C_p\wr C_2$ of degree $2p$. Let $\Gamma$ be a connected $G$-vertex-transitive and $G$-edge-transitive graph and let $v$ be a vertex of $\Gamma$. We show that if the permutation group induced by the vertex-stabiliser $G_v$ on the neighbourhood $\Gamma (v)$ is isomorphic to $L$ then either $|V(\Gamma )|\ge p|G_v|{\log }_p\left(|G_v|/2\right)$, or $|V(\Gamma )|$ is bounded by a constant depending only on $p$, or $\Gamma$ is a very-well understood graph. This generalises a few recent results.

References in corpus (1)

• Bounding the order of the vertex-stabiliser in 3-valent vertex-transitive and 4-valent arc-transitive graphs

Cited by in corpus (3)

• Recipes for Edge-Transitive Tetravalent Graphs
• On the maximum order of nilpotent transitive permutation groups
• Semiregular automorphisms of cubic vertex-transitive graphs