Vertex-transitive graph

In the mathematical field of graph theory, a vertex-transitive graph is a graph G in which, given any two vertices v₁ and v₂ of G, there is some automorphism

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

such that

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

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