Vertex-transitive graph

In the mathematical field of graph theory, an automorphism is a permutation of the vertices such that edges are mapped to edges and non-edges are mapped to non-edges.<ref name=Godsil01/> A graph is a vertex-transitive graph if, given any two vertices Template:Math and Template:Math of Template:Mvar, there is an automorphism Template:Math such that

<math>f(v_1) = v_2.\ </math>

In other words, a graph is vertex-transitive if its automorphism group acts transitively on its vertices.<ref name=Godsil01>Template:Citation.</ref> 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

File:Tuncated tetrahedral graph.png

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.<ref>Template:Citation.</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 graphs of edge-transitive non-bipartite graphs with odd vertex degrees.<ref>Template:Citation. Lauri and Scapelleto credit this construction to Mark Watkins.</ref>

Properties

The edge-connectivity of a connected vertex-transitive graph is equal to the degree d, while the vertex-connectivity will be at least 2(d + 1)/3.<ref name=Godsil01/> 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.<ref>Template:Citation</ref>

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.<ref>Template:Citation.</ref> In 2005, Eskin, Fisher, and Whyte confirmed the counterexample.<ref>Template:Cite arXiv.</ref>

See also

• Edge-transitive graph
• Lovász conjecture
• Semi-symmetric graph
• Zero-symmetric graph

References

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External links

• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Vertex-TransitiveGraph%7CVertex-TransitiveGraph.html}} |title = Vertex-transitive graph |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

• A census of small connected cubic vertex-transitive graphs. Primož Potočnik, Pablo Spiga, Gabriel Verret, 2012.
• Vertex-transitive Graphs On Fewer Than 48 Vertices. Gordon Royle and Derek Holt, 2020.