Automorphism

Automorphism — автоморфизм (ор)графа.

1. For an undirected graph, see Isomorphic graphs.

2. For a directed graph, automorphism is a permutation [math]\displaystyle{ \,\alpha }[/math] of [math]\displaystyle{ \,V(G) }[/math] such that the number of [math]\displaystyle{ \,(x,y) }[/math]-edges is the same as the number of [math]\displaystyle{ (\,\alpha(x), \alpha(y)) }[/math]-edges [math]\displaystyle{ (x,y \in V(G)) }[/math]. We also speak of the automorphism of a graph [math]\displaystyle{ \,G }[/math] with colored edges. This means a permutation [math]\displaystyle{ \,\alpha }[/math] such that the number of [math]\displaystyle{ \,(x,y) }[/math]-edges is the same as the number of [math]\displaystyle{ (\,\alpha(x), \alpha(y)) }[/math]-edges with any given color.

The set of all automorphisms of a (di)graph forms a permutation group [math]\displaystyle{ \,A(G) }[/math].