Vertex-transitive graph

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

f:V(G)\to V(G)\

such that

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

In other words, a graph is vertex-transitive if its automorphism group acts transitively on 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).

Graph families defined by their automorphisms
distance-transitive	→	distance-regular	←	strongly regular
↓
symmetric (arc-transitive)	←	t-transitive, $t \geq 2$		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