Vertex-transitive graph

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

${\displaystyle f:V(G)\rightarrow V(G)\ }$

such that

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

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:

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

1. ^ Godsil, Chris; Royle, Gordon (2001), Algebraic Graph Theory, Graduate Texts in Mathematics, 207, New York: Springer-Verlag.
2. ^ Potočnik P., Spiga P. & Verret G. (2013), "Cubic vertex-transitive graphs on up to 1280 vertices", Journal of Symbolic Computation, 50: 465–477, doi:10.1016/j.jsc.2012.09.002.
3. ^ 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.
4. ^ Godsil, C. & Royle, G. (2001), Algebraic Graph Theory, Springer Verlag
5. ^ Babai, L. (1996), Technical Report TR-94-10, University of Chicago[1]
6. ^ 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.
7. ^ Eskin, Alex; Fisher, David; Whyte, Kevin (2005). "Quasi-isometries and rigidity of solvable groups". arXiv:..