On the stabilisers of locally 2-transitive graphs

TL;DR

This paper classifies the possible stabilizer triples in connected locally 2-transitive graphs when vertex stabilizers act faithfully on their neighborhoods, revealing exactly 16 such configurations except when they are isomorphic.

Contribution

It provides a classification of stabilizer triples in locally 2-transitive graphs under faithful action conditions, a significant step in symmetrical graph theory.

Findings

Exactly 16 stabilizer triples when vertex stabilizers are not isomorphic.

Complete classification under the faithful action assumption.

Identification of the exceptional case where stabilizers are isomorphic.

Abstract

For a connected locally $(G,s$)-arc-transitive graph $\Gamma$ with $s\geqslant 2$ and an edge ${v,w}$, determining the amalgam $(G_v,G_w,G_{vw})$ is a fundamental problem in the area of symmetrical graph theory, but it is very difficult. In this paper, we give a classification of $(G_v,G_w,G_{vw})$ in the case where the vertex stabilisers $G_v$ and $G_w$ are faithful on their neighbourhoods, which shows that except for the case $G_v\cong G_w$, there are exactly 16 such triples.