Graph automorphism

An isomorphic mapping of a graph onto itself (cf. Graph isomorphism). The set of all automorphisms of a given graph forms a group with respect to the operation of composition of automorphisms. The automorphisms of a graph $G$ generate a group $\Gamma(G)$ of permutations of vertices, which is called the group (or vertex group) of $G$, and a group of edge permutations $\Gamma_1(G)$, called the edge group of $G$. The edge group and vertex group of a graph $G$ without loops and multiple edges are isomorphic if and only if $G$ contains not more than one isolated vertex and if none of its connected components is an isolated edge. For each finite group $F$ there exists a graph whose automorphism group is isomorphic to $F$. There also exist permutation groups on a set of $n$ elements which are not the vertex group of any graph with $n$ vertices. Various types and measures of symmetry of a graph can be related to its automorphisms. A graph with no automorphisms other than the identical one is said to be asymmetric. If $n\to\infty$, almost all graphs with $n$ vertices are asymmetric.

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The fact that for a finite group $F$ there is a graph with automorphism group $F$ is due to R. Frucht [a3]. A good reference for this and other algebraic aspects of graph theory is [a1]. A related reference is [a2].

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