Definition
Suppose that is a group and is a generating set. The Cayley 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 are joined 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 (i.e. ) 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) if and only if .
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