An automorphism of a graph is a graph isomorphism with itself, i.e., a mapping from the vertices of the given graph G back to vertices of G such that the resulting graph is isomorphic with G. The set of automorphisms defines a permutation group known as the graph's automorphism group. For every group Γ, there exists a graph whose automorphism group is isomorphic to Γ. The automorphism groups of a graph characterize its symmetries, and are therefore very useful in determining certain of its properties.

automorphism group | cyclic group graph | edge automorphism | edge automorphism group | edge-transitive graph | Frucht graph | graph isomorphism | isomorphic graphs | vertex-transitive graph