Given a group G and a subset S such that the identity is in S and such that
S^{-1} = S, the Cayley graph associated with the pair (G,S) is
a directed graph which has as vertices the elements of G and for which
(g_{1}, g_{2}) is an edge if g_{1}g_{2}^{-1} is in S.

They graph may depend on the choice of the subset S. Cayley Graphs are always vertex-transitive.

The Cayley graph is connected if and only if the set S generates the group G.

When instead of a subset S we take a subgroup H of finite index in G we obtain a graph that is at the basis of coset enumeration or Todd-Coxeter process.