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:
“If I could only live at the pitch that is near madness
When everything is as it was in my childhood
Violent, vivid, and of infinite possibility:
That the sun and the moon broke over my head.”
—Richard Eberhart (b. 1904)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)