# Vertex-transitive graph

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

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]