# On edge-primitive and 2-arc-transitive graphs

## TL;DR

This paper investigates the structure of edge-primitive, 2-arc-transitive graphs, proving they have almost simple automorphism groups unless they are cycles or complete bipartite graphs, and provides examples with specific symmetry properties.

## Contribution

It establishes a classification result for finite 2-arc-transitive edge-primitive graphs and presents explicit examples with high symmetry.

## Key findings

- Finite 2-arc-transitive edge-primitive graphs have almost simple automorphism groups.
- Examples of 3-arc-transitive graphs with faithful vertex-stabilizers are provided.
- The classification excludes cycles and complete bipartite graphs.

## Abstract

A graph is edge-primitive if its automorphism group acts primitively on the edge set. In this short paper, we prove that a finite 2-arc-transitive edge-primitive graph has almost simple automorphism group if it is neither a cycle nor a complete bipartite graph. We also present two examples of such graphs, which are 3-arc-transitive and have faithful vertex-stabilizers.