Cayley graph Guide, Meaning , Facts, Information and Description

In mathematics, a Cayley graph is a graph which can be associated to a group; it is a central tool in combinatorial and geometric group theory.

Strictly speaking the Cayley graph depends not only on the group, but also on a choice of generators for the group. Let G be a group, and let S be a set of generators for G (usually assumed to be symmetric, i.e. g∈S iff g^-1∈S ). Then the Cayley graph of G with respect to S is a graph which has as vertices the elements of G; two vertices g₁ and g₂ are connected by an edge if and only if g₁^-1g₂ is an element of S.

G acts on itself by multiplication on the left, and this action preserves the graph structure. Since the action of G on itself is transitive, any Cayley graph is vertex-transitive.

If one takes the vertices to instead be right cosets of a fixed subgroup H, one obtains a related construction, the Schreier coset graph, which is at the basis of coset enumeration or the Todd-Coxeter process.

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