Edge-transitive graph

In the mathematical field of graph theory, an edge-transitive graph is a graph "G" such that, given any two edges "e"1 and "e"2 of "G", there is an 
automorphism of "G" that maps "e"1 to "e"2.
In other words, a graph is edge-transitive if its automorphism group acts transitively on its edges.
Examples and properties.
Edge-transitive graphs include any complete bipartite graph formula_1, and any symmetric graph, such as the vertices and edges of the cube. Symmetric graphs are also vertex-transitive (if they are connected), but in general edge-transitive graphs need not be vertex-transitive. The Gray graph is an example of a graph which is edge-transitive but not vertex-transitive. All such graphs are bipartite, and hence can be colored with only two colors.
An edge-transitive graph that is also regular, but not vertex-transitive, is called semi-symmetric. The Gray graph again provides an example.
Every edge-transitive graph that is not vertex-transitive must be bipartite and either semi-symmetric or biregular.
The vertex connectivity of an edge-transitive graph always equals its minimum degree.
Marston Conder has compiled a Complete list of all connected edge-transitive graphs on up to 47 vertices and a Complete list of all connected edge-transitive bipartite graphs on up to 63 vertices.