Vertex-transitive graph

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Graph families defined by their automorphisms
distance-transitive	$\rightarrow$	distance-regular	$\leftarrow$	strongly regular
$\downarrow$
symmetric (arc-transitive)	$\leftarrow$	t-transitive, t ≥ 2
$\downarrow$ _{(if connected)}
vertex- and edge-transitive	$\rightarrow$	edge-transitive and regular	$\rightarrow$	edge-transitive
$\downarrow$		$\downarrow$		$\downarrow$
vertex-transitive	$\rightarrow$	regular	$\rightarrow$	_{(if bipartite)} biregular
$\uparrow$
Cayley graph		skew-symmetric		asymmetric

In the mathematical field of graph theory, a vertex-transitive graph is a graph G such that, given any two vertices v₁ and v₂ of G, there is some automorphism

$f:V(G)\rightarrow V(G)\$

such that

$f(v_{1})=v_{2}.\$

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

The edges of the truncated tetrahedron form a vertex-transitive graph (also a Cayley graph) which is not symmetric.

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. and Verret G. (2012), Cubic vertex-transitive graphs on up to 1280 vertices, Journal of Symbolic Computation .
↑ 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. and Royle, G. (2001), Algebraic Graph Theory, Springer Verlag
↑ Babai, L. (1996), Technical Report TR-94-10, University of Chicago
↑ Diestel, Reinhard; Leader, Imre (2001), "A conjecture concerning a limit of non-Cayley graphs", 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

A census of small connected cubic vertex-transitive graphs . Primož Potočnik, Pablo Spiga, Gabriel Verret, 2012.

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