Vertex-transitive graph

In mathematics, 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)" → "V(G)"

such that

:"f" (v₁) = v₂.

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 vertex-transitive graph is regular. Every arc-transitive graph without isolated vertices is also vertex-transitive.

Finite examples

* Heawood graph
* Kneser graph and its complement Johnson graph
** Petersen graph
* finite Cayley graphs
** circulant graphs
*** Complete graph
*** Cycle graph
*** Complete bipartite graph K_{n,n}
* graphs of Platonic solids and Archimedean solids

Infinite examples

* infinite path (infinite in both directions)
* infinite regular tree, 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

Infinite vertex-transitive graphs

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

References

Wikimedia Foundation. 2010.

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Vertex-transitive graph

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