Implicit function theorem: Difference between revisions

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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]]

Latest revision as of 22:19, 29 December 2014

Have you been thinking "how do I speed up my computer" lately? Well chances are should you are reading this article; then you may be experiencing one of various computer issues which thousands of people discover which they face regularly.

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You should furthermore see to it which it is truly easy to download and install. We must avoid those goods that usually need we a rather complicated set of instructions. Furthermore, you need to no longer want any additional system requirements.

The System File Checker (SFC) may aid in resolving error 1721 because it, by its nature, scans the system files for corruption and replaces them with their authentic versions. This requires we to have the Windows Installation DVD ROM for continuing.

All of these problems is conveniently solved by the clean registry. Installing our registry cleaner allows we to use your PC without worries behind. You may able to utilize we system without being afraid which it's going to crash inside the center. Our registry cleaner can fix a host of errors on your PC, identifying missing, invalid or corrupt settings in the registry.

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