In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity.
Formally, an automorphism of a graph G = (V,E) is a permutation σ of the vertex set V, such that the pair of vertices (u,v) form an edge if and only if the pair (σ(u),σ(v)) also form an edge. That is, it is a graph isomorphism from G to itself. Automorphisms may be defined in this way both for directed graphs and for undirected graphs. The composition of two automorphisms is another automorphism, and the set of automorphisms of a given graph, under the composition operation, forms a group, the automorphism group of the graph. In the opposite direction, by Frucht's theorem, all groups can be represented as the automorphism group of a connected graph – indeed, of a cubic graph.
Other articles related to "graph automorphism, graphs, automorphisms, graph, automorphism":
... Several families of graphs are defined by having certain types of automorphisms An asymmetric graph is an undirected graph without any nontrivial automorphisms ... A vertex-transitive graph is an undirected graph in which every vertex may be mapped by an automorphism into any other vertex ... An edge-transitive graph is an undirected graph in which every edge may be mapped by an automorphism into any other edge ...
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