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 $Γ$ be a connected $G$-vertex-transitive and $G$-edge-transitive graph and let $v$ be a vertex of $Γ$. We show that if the permutation group induced by the vertex-stabiliser $G_v$ on the neighbourhood $Γ(v)$ is isomorphic to $L$ then either $|V(Γ)|\geq p|G_v|\log_p\left(|G_v|/2\right)$, or $|V(Γ)|$ is bounded by a constant depending only on $p$, or $Γ$ is a very-well understood graph. This generalises a few recent results.