Edge-Transitive Graphs

TL;DR

This paper provides a complete classification of small connected edge-transitive graphs, introduces a construction for infinite bipartite edge-transitive graphs, and explores conditions for edge transitivity in specific graph classes.

Contribution

It offers a comprehensive classification of small edge-transitive graphs, constructs infinite bipartite families, and analyzes transitivity conditions in biregular and circulant graphs.

Findings

Complete classification of connected edge-transitive graphs up to 20 vertices.

Construction of an infinite family of bipartite edge-transitive graphs.

Conditions for edge transitivity in circulant and biregular graphs.

Abstract

A graph is said to be edge-transitive if its automorphism group acts transitively on its edges. It is known that edge-transitive graphs are either vertex-transitive or bipartite. In this paper we present a complete classification of all connected edge-transitive graphs on less than or equal to $20$ vertices. We then present a construction for an infinite family of edge-transitive bipartite graphs, and use this construction to show that there exists a non-trivial bipartite subgraph of $K_{m,n}$ that is connected and edge-transitive whenever $gcd(m,n)>2$. Additionally, we investigate necessary and sufficient conditions for edge transitivity of connected $(r,2)$ biregular subgraphs of $K_{m,n}$, as well as for uniqueness, and use these results to address the case of $gcd(m,n)=2$. We then present infinite families of edge-transitive graphs among vertex-transitive graphs, including several classes of circulant graphs. In particular, we present necessary conditions and sufficient conditions for edge-transitivity of certain circulant graphs.