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Cayley Graphs & Cayley Maps

Cayley Graphs
Cayley Graphs are a way of representing groups. Let $G$ be a group with generating set $S$ . The Cayley Graph, $C_G(G,S)$ , is a graph where all members of $G$ are vertices, and two vertices, $v$ and $w$ , are adjacent if $w=vs$ or $v=ws$ for some $s \in S$ .

**Cayley Graph $C_G(Z_{10},\{1,3,5\})$ , drawn two ways.** The red lines represent the vertices that are connected by the 1 in the generating set, the blue lines represent the vertices connected by the 3, and the green lines represent the vertices connected by the 5. Notice that $C_G(Z_{10},{1,3,5})$ is the same as $K_{5,5}$ .

**Cayley Graph $C_G(Z_{10},\{1,3,5\})$ , drawn two ways.** The red lines represent the vertices that are connected by the 1 in the generating set, the blue lines represent the vertices connected by the 3, and the green lines represent the vertices connected by the 5. Notice that $C_G(Z_{10},{1,3,5})$ is the same as $K_{5,5}$ .

Cayley Maps
Let $S'$ be a closed generating set of $G$ . The Cayley Map $C_M(G,\rho)$ is an embedding of $C_G(G,S')$ onto an orientable surface, where $\rho=(x_0,\dots, x_{n-1})$ is a rotation (cyclic permutation) of $S'$ .

Cayley Map $C_M(G,(x_0,\dots, x_{n-1}))$ where $a \in G$ . Notice that the elements in the rotation are assigned to the branches of the Cayley Map in counterclockwise order

Cayley Map $C_M(G,(x_0,\dots, x_{n-1}))$ where $a \in G$ . Notice that the elements in the rotation are assigned to the branches of the Cayley Map in counterclockwise order

Graph Embeddings

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Cayley Graphs & Cayley Maps