Vertex-transitive Graph - Infinite Examples

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.

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