# Vertex-transitive graph

In the mathematical field of graph theory, a **vertex-transitive graph** is a graph *G* such that, given any two vertices v_{1} and v_{2} of *G*, there is some automorphism

Graph families defined by their automorphisms | ||||
---|---|---|---|---|

distance-transitive | → | distance-regular | ← | strongly regular |

↓ | ||||

symmetric (arc-transitive) | ← | t-transitive, t ≥ 2 |
skew-symmetric | |

↓ | ||||

_{(if connected)}vertex- and edge-transitive |
→ | edge-transitive and regular | → | edge-transitive |

↓ | ↓ | ↓ | ||

vertex-transitive | → | regular | → | _{(if bipartite)}biregular |

↑ | ||||

Cayley graph | ← | zero-symmetric | asymmetric |

such that

In other words, a graph is vertex-transitive if its automorphism group acts transitively upon its vertices.[1] 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. 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

Finite vertex-transitive graphs include the symmetric graphs (such as the Petersen graph, the Heawood graph and the vertices and edges of the Platonic solids). The finite Cayley graphs (such as cube-connected cycles) are also vertex-transitive, as are the vertices and edges of the Archimedean solids (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.[2]

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 graphs of edge-transitive non-bipartite graphs with odd vertex degrees.[3]

## Properties

The edge-connectivity of a vertex-transitive graph is equal to the degree *d*, while the vertex-connectivity will be at least 2(*d* + 1)/3.[4]
If the degree is 4 or less, or the graph is also edge-transitive, or the graph is a minimal Cayley graph, then the vertex-connectivity will also be equal to *d*.[5]

## Infinite examples

Infinite vertex-transitive graphs include:

- infinite paths (infinite in both directions)
- infinite regular trees, e.g. the Cayley graph of the free group
- graphs of uniform tessellations (see a complete list of planar tessellations), including all tilings by regular polygons
- infinite Cayley graphs
- the Rado graph

Two countable vertex-transitive graphs are called quasi-isometric if the ratio of their distance functions 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 Diestel and Leader in 2001.[6] In 2005, Eskin, Fisher, and Whyte confirmed the counterexample.[7]

## References

- Godsil, Chris; Royle, Gordon (2001),
*Algebraic Graph Theory*, Graduate Texts in Mathematics,**207**, New York: Springer-Verlag. - Potočnik P., Spiga P. & Verret G. (2013), "Cubic vertex-transitive graphs on up to 1280 vertices",
*Journal of Symbolic Computation*,**50**: 465–477, arXiv:1201.5317, doi:10.1016/j.jsc.2012.09.002. - Lauri, Josef; Scapellato, Raffaele (2003),
*Topics in graph automorphisms and reconstruction*, London Mathematical Society Student Texts,**54**, Cambridge: Cambridge University Press, p. 44, ISBN 0-521-82151-7, MR 1971819. Lauri and Scapelleto credit this construction to Mark Watkins. - Godsil, C. & Royle, G. (2001),
*Algebraic Graph Theory*, Springer Verlag - Babai, L. (1996),
*Technical Report TR-94-10*, University of Chicago Archived 2010-06-11 at the Wayback Machine - Diestel, Reinhard; Leader, Imre (2001), "A conjecture concerning a limit of non-Cayley graphs" (PDF),
*Journal of Algebraic Combinatorics*,**14**(1): 17–25, doi:10.1023/A:1011257718029. - Eskin, Alex; Fisher, David; Whyte, Kevin (2005). "Quasi-isometries and rigidity of solvable groups". arXiv:math.GR/0511647..

## External links

- Weisstein, Eric W. "Vertex-transitive graph".
*MathWorld*. - A census of small connected cubic vertex-transitive graphs . Primož Potočnik, Pablo Spiga, Gabriel Verret, 2012.