On the order of arc-stabilisers in arc-transitive graphs with prescribed local group

TL;DR

This paper investigates bounds on the size of arc-stabilizers in connected arc-transitive graphs based on local group actions and graph order, contributing to understanding symmetry constraints.

Contribution

It provides new bounds on arc-stabilizer orders in arc-transitive graphs using local group actions and graph order parameters.

Findings

Derived bounds on $|G_{uv}|$ in terms of local group $L$ and graph order

Established relationships between local symmetry groups and global automorphism structure

Enhanced understanding of symmetry constraints in arc-transitive graphs

Abstract

Let $\Gamma$ be a connected $G$-arc-transitive graph, let $uv$ be an arc of $\Gamma$ and let $L$ be the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $\Gamma(v)$. We study the problem of bounding $|G_{uv}|$ in terms of $L$ and the order of $\Gamma$.