Vertex-transitive graph

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

In the mathematical field of graph theory, a vertex-transitive graph is a graph G such that, given any two vertices v₁ and v₂ of G, there is some automorphism

$f:V(G) \rightarrow V(G)\$

such that

$f(v_1) = v_2.\$

In other words, a graph is vertex-transitive if its automorphism group acts transitively upon its vertices. A graph is vertex-transitive if and only if its graph complement is, since the group actions are identical.

Every symmetric graph without isolated vertices is vertex-transitive, and every vertex-transitive graph is regular. However, not all vertex-transitive graphs are symmetric (for example, the edges of the truncated tetrahedron), and not all regular graphs are vertex-transitive (for example, the Frucht graph).

Finite examples

The edges of the truncated tetrahedron form a vertex-transitive graph (also a Cayley graph) which is not symmetric.

Finite vertex-transitive graphs include the symmetric graphs (such as the Petersen graph, the Heawood graph and the vertices and edges of the Platonic solids). The finite Cayley graphs (such as cube-connected cycles) are also vertex-transitive, as are the vertices and edges of the Archimedean solids (though only two of these are symmetric).

Properties

The edge-connectivity of a vertex-transitive graph is equal to the degree d, while the vertex-connectivity will be at least 2(d+1)/3. If the degree is 4 or less, or the graph is also edge-transitive, or the graph is a minimal Cayley graph, then the vertex-connectivity will also be equal to d.

Infinite examples

Infinite vertex-transitive graphs include:

• infinite paths (infinite in both directions)
• infinite regular trees, e.g. the Cayley graph of the free group
• graphs of uniform tessellations (see a complete list of planar tessellations), including all tilings by regular polygons
• infinite Cayley graphs
• the Rado graph

Two countable vertex-transitive graphs are called quasi-isometric if the ratio of their distance functions is bounded from below and from above. A well known conjecture states that every infinite vertex-transitive graph is quasi-isometric to a Cayley graph. A counterexample has been proposed by Diestel and Leader. Most recently, Eskin, Fisher, and Whyte confirmed the counterexample.

See also

• Edge-transitive graph
• Lovász conjecture
• Semi-symmetric graph