Edge-transitive graph

In the mathematical field of graph theory, an edge-transitive graph is a graph G such that, given any two edges e1 and e2 of G, there is an automorphism of G that maps e1 to e2. In the mathematical field of graph theory, an edge-transitive graph is a graph G such that, given any two edges e1 and e2 of G, there is an automorphism of G that maps e1 to e2. In other words, a graph is edge-transitive if its automorphism group acts transitively upon its edges. Edge-transitive graphs include any complete bipartite graph K m , n {displaystyle K_{m,n}} , 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.