Edge-transitive graphs and combinatorial designs

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. We present a complete classification of all connected edge-transitive graphs on less than or equal to 20 vertices. We investigate biregular bipartite edge-transitive graphs and present connections to combinatorial designs, and we show that the Cartesian products of complements of complete graphs give an additional family of edge-transitive graphs.