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 |
In the mathematical field of graph theory, a vertex-transitive graph is a graph G in which, given any two vertices v_{1} and v_{2} of G, there is some automorphism
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]}
See also
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[1] 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.