Edge-transitive graph explained

In the mathematical field of graph theory, an edge-transitive graph is a graph such that, given any two edges and of, there is an automorphism of that maps to .^[1]

In other words, a graph is edge-transitive if its automorphism group acts transitively on its edges.

Examples and properties

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 ...

Edge-transitive graphs include all 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. Every connected edge-transitive graph that is not vertex-transitive must be bipartite, (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

K _{m,n} where m ≠ n, which includes the star graphs K _{1,n} . 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 K_m,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 K_3,6, K_4,6 and K_5,10 but not K_4,10. 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.

The vertex connectivity of an edge-transitive graph always equals its minimum degree.

See also

• Edge-transitive (in geometry)

Notes and References

1. Book: Biggs, Norman . Algebraic Graph Theory . 2nd . Cambridge . Cambridge University Press . 1993 . 118 . 0-521-45897-8.
2. .
3. Newman . Heather A. . Miranda . Hector . Narayan . Darren A . Edge-transitive graphs and combinatorial designs . Involve: A Journal of Mathematics . 2019 . 12 . 8 . 1329–1341 . 10.2140/involve.2019.12.1329 . 1709.04750 . 119686233 .