Graphs

Definition 3.1   The right regular representation of a group $G$ is the homomorphism $\rho\colon G\to \Sym(G)$ given by $g\mapsto \rho_g$, where $\rho_g$ is the permutation that acts as $x^{\rho_g}=xg$. The image (which is isomorphic to $G$) is denoted $R(G)$.

Corollary 3.2 (Cayley)   Every group is isomorphic to a permutation group.

Definition 3.3   If $G$ is generated by a set $S$, then the Cayley Color Diagram $\Gamma_c(G,S)$ is the colored digraph on the vertex set $G$ with edges pointing from $g$ to $sg$ labeled (colored) by $s$ for all $g\in G$ and $s\in S$. Automorphisms of such a colored digraph preserve the colors by definition.