Editing Petersen graph (section)

== Symmetries ==
The Petersen graph is [[strongly regular graph|strongly regular]] (with signature srg(10,3,0,1)). It is also [[symmetric graph|symmetric]], meaning that it is [[edge-transitive graph|edge transitive]] and [[vertex-transitive graph|vertex transitive]]. More strongly, it is 3-arc-transitive: every directed three-edge path in the Petersen graph can be transformed into every other such path by a symmetry of the graph.<ref>{{citation| first=László| last=Babai| author-link=László Babai|contribution=Automorphism groups, isomorphism, reconstruction|id=Corollary 1.8|title=Handbook of Combinatorics| url=http://www.cs.uchicago.edu/files/tr_authentic/TR-94-10.ps|pages=1447–1540|editor1-first=Ronald L.|editor1-last=Graham|editor1-link=Ronald Graham| editor2-first=Martin |editor2-last=Grötschel|editor2-link=Martin Grötschel|editor3-first=László|editor3-last=Lovász|editor3-link=László Lovász| volume =I |publisher=North-Holland|year=1995| archive-url= https://web.archive.org/web/20100611212234/http://www.cs.uchicago.edu/files/tr_authentic/TR-94-10.ps|archive-date=2010-06-11}}.</ref>
It is one of only 13 cubic [[distance-regular graph]]s.<ref name=foster/>

The [[automorphism group]] of the Petersen graph is the [[symmetric group]] <math>S_5</math>; the action of <math>S_5</math> on the Petersen graph follows from its construction as a [[Kneser graph]]. The Petersen graph is a [[core (graph theory)|core]]: every [[graph homomorphism|homomorphism]] of the Petersen graph to itself is an [[graph automorphism|automorphism]].<ref>{{citation|last=Cameron|first=Peter J.|editor1-last=Beineke|editor1-first=Lowell W.|editor2-last=Wilson|editor2-first=Robin J.|contribution=Automorphisms of graphs|doi=10.1017/CBO9780511529993|isbn=0-521-80197-4|mr=2125091|pages=135–153|publisher=Cambridge University Press, Cambridge|series=Encyclopedia of Mathematics and its Applications|title=Topics in Algebraic Graph Theory|volume=102|year=2004}}; see in particular [https://books.google.com/books?id=z2K26gZLC1MC&pg=PA153 p. 153]</ref> As shown in the figures, the drawings of the Petersen graph may exhibit five-way or three-way symmetry, but it is not possible to draw the Petersen graph in the plane in such a way that the drawing exhibits the full symmetry group of the graph.

Despite its high degree of symmetry, the Petersen graph is not a [[Cayley graph]]. It is the smallest vertex-transitive graph that is not a Cayley graph.{{efn|As stated, this assumes that Cayley graphs need not be connected. Some sources require Cayley graphs to be connected, making the two-vertex [[empty graph]] the smallest vertex-transitive non-Cayley graph; under the definition given by these sources, the Petersen graph is the smallest connected vertex-transitive graph that is not Cayley.}}