On the automorphism groups of graphs with twice prime valency

TL;DR

This paper classifies the automorphism groups of connected edge-transitive graphs with odd order and twice prime valency, showing they are either almost simple or affine primitive groups, with most socles acting transitively on edges.

Contribution

It provides a classification of automorphism groups for a specific class of graphs, extending understanding of their symmetry properties.

Findings

Automorphism groups are either almost simple or affine primitive.

Most socles of almost simple groups act transitively on edges.

Identifies two exceptions where socles do not act transitively.

Abstract

A graph is edge-transitive if its automorphism group acts transitively on the edge set. In this paper, we investigate the automorphism groups of edge-transitive graphs of odd order and twice prime valency. Let $\Gamma$ be a connected graph of odd order and twice prime valency, and let $G$ be a subgroup of the automorphism group of $\Ga$. In the case where $G$ acts transitively on the edges and quasiprimitively on the vertices of $\Ga$, we prove that either $G$ is almost simple or $G$ is a primitive group of affine type. If further $G$ is an almost simple primitive group then, with two exceptions, the socle of $G$ acts transitively on the edges of $\Gamma$.