Edge-Transitive 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 Km,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 Km,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.