A Cayley graph Cay(G, S) is a graph that encodes the structure of a group G with respect to a chosen generating set S.

Vertices correspond to elements of G. Edges are drawn from each element g to gs for every generator s ∈ S, with each generator given its own colour. The resulting graph reveals the group's multiplication structure visually.

Key properties:

• The graph is vertex-transitive — every vertex looks the same, reflecting the fact that group elements are all "equivalent" under right multiplication.

• The graph is connected if and only if S generates G.

• Each vertex has out-degree |S|.

• Different choices of generating set S produce different Cayley graphs, which is why the same group can appear as several distinct polyhedra above.

Cayley graphs are a powerful bridge between algebra and geometry: the symmetries of the graph reflect the algebraic structure of G, and geometric properties of the graph (such as diameter or girth) carry group-theoretic meaning.