Editing Vertex-transitive graph (section)

== Infinite examples ==
Infinite vertex-transitive graphs include:
* infinite [[Path (graph theory)|paths]] (infinite in both directions)
* infinite [[Regular graph|regular]] [[tree (graph theory)|trees]], e.g. the [[Cayley graph]] of the [[free group]]
* graphs of [[Uniform tiling|uniform tessellations]] (see a [[List of uniform planar tilings|complete list]] of planar [[tessellation]]s), including all [[Tiling by regular polygons|tilings by regular polygons]]
* infinite [[Cayley graph]]s
* the [[Rado graph]]

Two [[countable]] vertex-transitive graphs are called [[Glossary of Riemannian and metric geometry#Q|quasi-isometric]] if the ratio of their [[distance function]]s is bounded from below and from above.  A well known [[conjecture]] stated that every infinite vertex-transitive graph is quasi-isometric to a [[Cayley graph]].  A counterexample was proposed by [[Reinhard Diestel|Diestel]] and [[Imre Leader|Leader]] in 2001.<ref>{{citation|first1=Reinhard|last1=Diestel|first2=Imre|last2=Leader|authorlink2=Imre Leader|url=http://www.math.uni-hamburg.de/home/diestel/papers/Cayley.pdf|title=A conjecture concerning a limit of non-Cayley graphs|journal=Journal of Algebraic Combinatorics|volume=14|issue=1|year=2001|pages=17–25|doi=10.1023/A:1011257718029|s2cid=10927964|doi-access=free}}.</ref>  In 2005, Eskin, Fisher, and Whyte confirmed the counterexample.<ref>{{cite arXiv|first1=Alex|last1=Eskin|first2=David|last2=Fisher|first3=Kevin|last3=Whyte|eprint=math.GR/0511647 |title=Quasi-isometries and rigidity of solvable groups|year=2005}}.</ref>