The number of connected simple edge-transitive graphs on n vertices is 1, 1, 2, 3, 4, 6, 5, 8, 9, 13, 7, 19, 10, 16, 25, 26, 12, 28 ... Шаблон:OEIS
Edge-transitive graphs include all symmetric graph, such as the vertices and edges of the cube.[1] Symmetric graphs are also vertex-transitive (if they are connected), but in general edge-transitive graphs need not be vertex-transitive. Every connected edge-transitive graph that is not vertex-transitive must be bipartite,[1] (and hence can be colored with only two colors), and either semi-symmetric or biregular.[2]
Examples of edge but not vertex transitive graphs include the complete bipartite graphs <math>K_{m,n}</math> where m ≠ n, which includes the star graphs <math>K_{1,n}</math>. For graphs on n vertices, there are (n-1)/2 such graphs for odd n and (n-2) for even n.
Additional edge transitive graphs which are not symmetric can be formed as subgraphs of these complete bi-partite graphs in certain cases. Subgraphs of complete bipartite graphs Km,n exist when m and n share a factor greater than 2. When the greatest common factor is 2, subgraphs exist when 2n/m is even or if m=4 and n is an odd multiple of 6.[3] So edge transitive subgraphs exist for K3,6, K4,6 and K5,10 but not K4,10. An alternative construction for some edge transitive graphs is to add vertices to the midpoints of edges of a symmetric graph with v vertices and e edges, creating a bipartite graph with e vertices of order 2, and v of order 2e/v.
An edge-transitive graph that is also regular, but still not vertex-transitive, is called semi-symmetric. The Gray graph, a cubic graph on 54 vertices, is an example of a regular graph which is edge-transitive but not vertex-transitive. The Folkman graph, a quartic graph on 20 vertices is the smallest such graph.
