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Definition (Cayley Graph Of A Group)
Let $G$ be a group that is generated by a set $S$. The Cayley graph of $G$ generated by $S$, which we'll write as $(G,S)$, is a colored directed graph that satisfies the the following three properties:
- The vertex set is $G$. Each vertex corresponds to an element of the group.
- Each element $s \in S$ is assigned a unique color which we'll denote by $c_s$.
- For each color $c_s$, and each vertex $g$, we draw the colored arrow $(g ,s g)$.
Most of the time we assume that $S$ does not contain the identity. However, if it does contain the identity, then we just draw a colored loop $(g,g)$ at each vertex.
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