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.
Read more about this topic: Vertex-transitive Graph
Famous quotes containing the words infinite and/or examples:
“Not till we are lost, in other words not till we have lost the world, do we begin to find ourselves, and realize where we are and the infinite extent of our relations.”
—Henry David Thoreau (1817–1862)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)