Edge-transitive core-free Nest graphs

TL;DR

This paper classifies edge-transitive, core-free Nest graphs with specific symmetry properties, identifying them as certain well-known graphs and a particular cover of K_{3,3}.

Contribution

It provides a complete classification of edge-transitive, core-free Nest graphs, linking them to known graphs and a specific normal 2-cover.

Findings

Edge-transitive, core-free Nest graphs are isomorphic to known graphs or a specific cover.

Identifies the complement of the Petersen graph, H(2,4), Shrikhande graph, and a Z_2^4 cover of K_{3,3}.

Provides structural characterization of these graphs.

Abstract

A finite simple graph $\Gamma$ is called a Nest graph if it is regular of valency $6$ and admits an automorphism $\rho$ with two orbits of the same length such that at least one of the subgraphs induced by these orbits is a cycle. We say that $\Gamma$ is core-free if no non-trivial subgroup of the group generated by $\rho$ is normal in $\mathrm{Aut}(\Gamma)$. In this paper, we show that, if $\Gamma$ is edge-transitive and core-free, then it is isomorphic to one of the following graphs: the complement of the Petersen graph, the Hamming graph $H(2,4)$, the Shrikhande graph and a certain normal $2$-cover of $K_{3,3}$ by $\mathbb{Z}_2^4$.