**Graph Automorphism**

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.

Read more about Graph Automorphism: Computational Complexity, Algorithms, Software and Applications, Symmetry Display, Graph Families Defined By Their Automorphisms

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**Graph Automorphism**- Graph Families Defined By Their Automorphisms

... 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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“When producers want to know what the public wants, they *graph* it as curves. When they want to tell the public what to get, they say it in curves.”

—Marshall McLuhan (1911–1980)