# Cayley graph

Cayley graph also know as Cayley colour graph, Cayley diagram, group diagram, or colour group  is a graph that encodes the abstract structure of a group

Definition Suppose that $G$ is a group and $S$ is a generating set. The Caylet graph $\Gamma=\Gamma(G,S)$ is a colored directed graph constructed as follows

• Each element $g$ of $G$ is assigned a vertex: the vertex set $V(\Gamma)$ of $\Gamma$ is identified with $G$.
• Each generator $s$ of $S$ is assigned a color $c_s$.
• For any $g\in G,s\in S$, the vertices corresponding to the elements $g$ and $gs$ by a directed edge of colour $c_s$. Thus the edge set $E(\Gamma)$ consists of pairs of the form $(g,gs)$, with $s\in S$ providing the color.

In geometric group theory, the set $S$ is usually assumed to be finite, symmetric and not containing the identity element of the group. In this case, the uncolored Cayley graph is an ordinary graph; its edges are not oriented and it does not contain loops (single-element cycles).