Locally arc-transitive graphs of valence $\{3,4\}$ with trivial edge kernel

TL;DR

This paper classifies certain highly symmetric graphs with vertices of valence 3 and 4, identifying key groups and enumerating all such graphs up to 350 vertices with specific automorphism properties.

Contribution

It introduces nineteen finitely presented groups that serve as quotients for all such graphs' automorphism groups, advancing the classification of locally arc-transitive graphs.

Findings

Nineteen finitely presented groups identified as universal quotients.

Complete enumeration of graphs with up to 350 vertices under specified symmetry conditions.

Characterization of automorphism groups with trivial edge kernel.

Abstract

In this paper we consider connected locally $G$-arc-transitive graphs with vertices of valence 3 and 4, such that the kernel $G_{uv}^{[1]}$ of the action of an edge-stabiliser on the neighourhood $\Gamma(u) \cup \Gamma(v)$ is trivial. We find nineteen finitely presented groups with the property that any such group $G$ is a quotient of one of these groups. As an application, we enumerate all connected locally arc-transitive graphs of valence ${3,4}$ on at most 350 vertices whose automorphism group contains a locally arc-transitive subgroup $G$ with $G_{uv}^{[1]} = 1$.