# Graph automorphism

## Mapping a graph onto itself without changing edge-vertex connectivity / From Wikipedia, the free encyclopedia

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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.^{[1]}^{[2]}