Editing Vertex-transitive graph (section)

== Finite examples ==
[[File:Tuncated tetrahedral graph.png|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). Potočnik, 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|author1=Potočnik P., Spiga P.  |author2=Verret G. |name-list-style=amp |journal=Journal of Symbolic Computation |volume = 50 | year = 2013|pages = 465–477|doi=10.1016/j.jsc.2012.09.002|arxiv=1201.5317|s2cid=26705221 }}.</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 [[parity (mathematics)|odd]] vertex degrees.<ref>{{citation
 | last1 = Lauri | first1 = Josef
 | last2 = Scapellato | first2 = Raffaele
 | isbn = 0-521-82151-7
 | mr = 1971819
 | page = 44
 | publisher = Cambridge University Press
 | series = London Mathematical Society Student Texts
 | title = Topics in graph automorphisms and reconstruction
 | url = https://books.google.com/books?id=hsymFm0E0uIC&pg=PA44
 | volume = 54
 | year = 2003}}. Lauri and Scapelleto credit this construction to Mark Watkins.</ref>