Edge-transitive graph

Graph families defined by their automorphisms
distance-transitive	→	distance-regular	←	strongly regular
↓
symmetric (arc-transitive)	←	t-transitive, t ≥ 2		skew-symmetric
↓
(if connected) vertex- and edge-transitive	→	edge-transitive and regular	→	edge-transitive
↓		↓		↓
vertex-transitive	→	regular	→	(if bipartite) biregular
↑
Cayley graph	←	zero-symmetric		asymmetric

This article is about graph theory. For edge transitivity in geometry, see Edge-transitive.

In mathematics, 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.

Examples and properties

• Any complete bipartite graph $K_{m,n}$ is edge-transitive.
• Any edge-transitive graph that is not vertex-transitive is bipartite.
• An edge-transitive graph that is regular but not vertex-transitive is called semi-symmetric.
• Any symmetric graph is edge-transitive.

See also

• Vertex-transitive graph
• Symmetric graph
• Edge-transitive (in geometry)

External links

• Weisstein, Eric W. "Edge-transitive graph". MathWorld.