A 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 is a group and is a generating set. The Caylet graph is a colored directed graph constructed as follows
- Each element of is assigned a vertex: the vertex set of is identified with .
- Each generator of is assigned a color .
- For any , the vertices corresponding to the elements and by a directed edge of colour . Thus the edge set consists of pairs of the form , with providing the color.
In geometric group theory, the set 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).