On finite edge-primitive and edge-quasiprimitive graphs

TL;DR

This paper systematically analyzes edge-primitive and edge-quasiprimitive graphs using the O'Nan--Scott Theorem, classifying possible group actions and providing numerous examples, including all such graphs with almost simple groups of socle PSL(2,q).

Contribution

It offers a comprehensive classification of edge-primitive and edge-quasiprimitive graphs based on group actions, extending understanding of their structure and examples.

Findings

Classified possible edge and vertex actions for such graphs.

Provided numerous explicit examples of edge-primitive graphs.

Determined all G-edge-primitive graphs for G with socle PSL(2,q).

Abstract

Many famous graphs are edge-primitive, for example, the Heawood graph, the Tutte--Coxeter graph and the Higman--Sims graph. In this paper we systematically analyse edge-primitive and edge-quasiprimitive graphs via the O'Nan--Scott Theorem to determine the possible edge and vertex actions of such graphs. Many interesting examples are given and we also determine all $G$-edge-primitive graphs for $G$ an almost simple group with socle $PSL(2,q)$.