Edge-transitive graph

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

Graph families defined by their automorphisms
distance-transitive	$\boldsymbol{\rightarrow}$	distance-regular	$\boldsymbol{\leftarrow}$	strongly regular
$\boldsymbol{\downarrow}$
symmetric (arc-transitive)	$\boldsymbol{\leftarrow}$	t-transitive, t ≥ 2		skew-symmetric
$\boldsymbol{\downarrow}$
(if connected) vertex- and edge-transitive	$\boldsymbol{\rightarrow}$	edge-transitive and regular	$\boldsymbol{\rightarrow}$	edge-transitive
$\boldsymbol{\downarrow}$		$\boldsymbol{\downarrow}$		$\boldsymbol{\downarrow}$
vertex-transitive	$\boldsymbol{\rightarrow}$	regular	$\boldsymbol{\rightarrow}$	(if bipartite) biregular
$\boldsymbol{\uparrow}$
Cayley graph	$\boldsymbol{\leftarrow}$	zero-symmetric		asymmetric

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.

Examples and properties

File:Gray graph 2COL.svg

The Gray graph is edge-transitive and regular, but not vertex-transitive.

Edge-transitive graphs include any complete bipartite graph $K_{m,n}$, 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.

See also

• Edge-transitive (in geometry)

External links

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