vertextransitivity

Vertex-transitivity is a property of graphs describing a high degree of symmetry. A graph G = (V, E) is vertex-transitive if its automorphism group acts transitively on the vertex set V; equivalently, for any two vertices u and v in V there exists an automorphism f of G with f(u) = v. An automorphism is a relabeling of the vertices that preserves adjacency and non-adjacency.

A key consequence is that vertex-transitive graphs are regular: every vertex has the same degree. Because all vertices are structurally indistinguishable, the degree sequence is uniform. However, a regular graph need not be vertex-transitive; uniform degree does not by itself guarantee a single orbit of the automorphism group on vertices.

Typical examples of vertex-transitive graphs include complete graphs K_n, cycle graphs C_n, and the vertex sets of hypercubes Q_d, as well as the Petersen graph. These graphs illustrate different scales and structures while maintaining vertex symmetry.

Related concepts distinguish levels of symmetry. If a graph is both vertex-transitive and edge-transitive, it is often called symmetric or arc-transitive, reflecting transitivity on ordered pairs of adjacent vertices. There exist vertex-transitive graphs that are not edge-transitive. Vertex-transitive graphs also frequently arise in the study of Archimedean solids and other structures where uniform vertex properties are desirable.

In summary, vertex-transitivity captures the idea that every vertex is interchangeable under the graph’s symmetries, a central notion in algebraic and geometric graph theory.