On basic $2$-arc-transitive graphs

TL;DR

This paper advances the understanding of basic 2-arc-transitive graphs by building on Praeger's group-theoretic characterizations and exploring their structural properties.

Contribution

It extends Praeger's work by providing new insights into the structure and classification of basic 2-arc-transitive graphs.

Findings

Connected 2-arc-transitive graphs are normal covers of basic 2-arc-transitive graphs.

Group-theoretic structures of basic 2-arc-transitive graphs are further characterized.

Enhanced understanding of the automorphism groups of these graphs.

Abstract

A connected graph $\Gamma=(V,E)$ of valency at least $3$ is called a basic $2$-arc-transitive graph if its full automorphism group has a subgroup $G$ with the following properties: (i) $G$ acts transitively on the set of $2$-arcs of $\Gamma$, and (ii) every minimal normal subgroup of $G$ has at most two orbits on $V$. In her papers [17,18], Praeger proved a connected $2$-arc-transitive graph of valency at least $3$ is a normal cover of some basic $2$-arc-transitive graph, and characterized the group-theoretic structures for basic $2$-arc-transitive graphs. Based on Praeger's theorems on $2$-arc-transitive graphs, this paper presents a further understanding on basic $2$-arc-transitive graphs.