Locally arc-transitive graphs of valence {3,4} 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 neighbourhood Γ(u)?Γ(v) is trivial. We find 19 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 $G_{uv}^{[1]} = 1$ .