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 $S$.