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{{Graph families defined by their automorphisms}}
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In the [[mathematics|mathematical]] field of [[graph theory]], a  '''vertex-transitive graph''' is a [[Graph (mathematics)|graph]] ''G'' such that, given any two vertices v<sub>1</sub> and v<sub>2</sub> of ''G'', there is some [[Graph automorphism|automorphism]]
 
:<math>f:V(G) \rightarrow V(G)\ </math>
 
such that  
 
:<math>f(v_1) = v_2.\ </math>
 
In other words, a graph is vertex-transitive if its automorphism group [[Group action|acts transitively]] upon its vertices.<ref>{{citation|first1=Chris|last1=Godsil|authorlink1=Chris Godsil|first2=Gordon|last2=Royle|authorlink2=Gordon Royle|title=Algebraic Graph Theory|series=Graduate Texts in Mathematics|volume=207|publisher=Springer-Verlag|location=New York|year=2001}}.</ref> A graph is vertex-transitive [[if and only if]] its [[graph complement]] is, since the group actions are identical.
 
Every [[symmetric graph]] without isolated vertices is vertex-transitive, and every vertex-transitive graph is [[Regular graph|regular]]. However, not all vertex-transitive graphs are symmetric (for example, the edges of the [[truncated tetrahedron]]), and not all regular graphs are vertex-transitive (for example, the [[Frucht graph]] and [[Tietze's graph]]).
 
== Finite examples ==
[[Image:TruncatedTetrahedron.gif|thumb|right|220px|The edges of the [[truncated tetrahedron]] form a vertex-transitive graph (also a [[Cayley graph]]) which is not [[symmetric graph|symmetric]].]]
Finite vertex-transitive graphs include the [[symmetric graph]]s (such as the [[Petersen graph]], the [[Heawood graph]] and the vertices and edges of the [[Platonic solid]]s).  The finite [[Cayley graph]]s (such as [[cube-connected cycles]]) are also vertex-transitive, as are the vertices and edges of the [[Archimedean solid]]s (though only two of these are symmetric). Poto&#269;nik, Spiga and Verret have constructed a census of all connected cubic vertex-transitive graphs on at most 1280 vertices.<ref>{{citation|title=Cubic vertex-transitive graphs on up to 1280 vertices|author=Poto&#269;nik P., Spiga P. and Verret G.|year=2012|publisher=Journal of Symbolic Computation}}.</ref>
 
Although every Cayley graph is vertex-transitive, there exist other vertex-transitive graphs that are not Cayley graphs. The most famous example is the Petersen graph, but others can be constructed including the [[line graph]]s of [[edge-transitive graph|edge-transitive]] non-[[bipartite graph|bipartite]] graphs with odd vertex degrees.<ref>{{citation
| last1 = Lauri | first1 = Josef
| last2 = Scapellato | first2 = Raffaele
| isbn = 0-521-82151-7
| location = Cambridge
| mr = 1971819
| page = 44
| publisher = Cambridge University Press
| series = London Mathematical Society Student Texts
| title = Topics in graph automorphisms and reconstruction
| url = http://books.google.com/books?id=hsymFm0E0uIC&pg=PA44
| volume = 54
| year = 2003}}. Lauri and Scapelleto credit this construction to Mark Watkins.</ref>
 
== Properties ==
The [[Connectivity (graph theory)|edge-connectivity]] of a vertex-transitive graph is equal to the [[regular graph|degree]] ''d'', while the [[Connectivity (graph theory)|vertex-connectivity]] will be at least 2(''d''+1)/3.<ref>{{Citation|title=Algebraic Graph Theory|author=Godsil, C. and Royle, G.|year=2001|publisher=Springer Verlag}}</ref>
If the degree is 4 or less, or the graph is also [[edge-transitive graph|edge-transitive]], or the graph is a minimal [[Cayley graph]], then the vertex-connectivity will also be equal to ''d''.<ref>{{Citation|title=Technical Report TR-94-10|author=Babai, L.|year=1996|publisher=University of Chicago}}[http://www.cs.uchicago.edu/files/tr_authentic/TR-94-10.ps]</ref>
 
== 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 tessellation]]s (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}}.</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>
 
== See also ==
* [[Edge-transitive graph]]
* [[Lovász conjecture]]
* [[Semi-symmetric graph]]
 
== References ==
<references/>
 
==External links==
*[http://www.matapp.unimib.it/~spiga/index.html A census of small connected cubic vertex-transitive graphs ]. Primo&#382; Poto&#269;nik, Pablo Spiga, Gabriel Verret, 2012.
 
[[Category:Graph families]]
[[Category:Algebraic graph theory]]
[[Category:Regular graphs]]

Revision as of 04:56, 1 March 2014

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