On edge-primitive 3-arc-transitive graphs

TL;DR

This paper advances the classification of highly symmetric graphs by analyzing edge-primitive 3-arc-transitive graphs with specific automorphism group structures, focusing on almost simple groups with alternating, sporadic, or classical socles.

Contribution

It provides a classification framework for edge-primitive 3-arc-transitive graphs with automorphism groups having particular almost simple structures, expanding understanding of their symmetry properties.

Findings

Classified graphs with automorphism groups having alternating or sporadic socles.

Analyzed graphs with classical group automorphisms and faithful vertex-stabilizers.

Established foundational results for further classification of symmetric graphs.

Abstract

This paper begins the classification of all edge-primitive 3-arc-transitive graphs by classifying all such graphs where the automorphism group is an almost simple group with socle an alternating or sporadic group, and all such graphs where the automorphism group is an almost simple classical group with a vertex-stabiliser acting faithfully on the set of neighbours.