Cayley graph of a group

Definition
Let $$G$$ be a group and $$S$$ be a generating set for $$G$$. The Cayley graph of $$G$$ with respect to $$S$$ is defined as follows:


 * The vertex set of the graph is $$G$$.
 * Given two distinct vertices $$g,h \in G$$, there is an edge joining $$g$$ to $$h$$ if and only if $$g^{-1}h$$ is in $$S \cup S^{-1}$$.

We typically consider the Cayley graph for a finitely generated group and a finite generating set of the group. Further, we can assume without loss of generality that $$S$$ is a symmetric subset of $$G$$ -- the inverse of any element of $$S$$ is also in $$S$$.