Cayley graph
Connection to group theory

Knowledge about the structure of the group can be obtained by studying the adjacency matrix of the graph and in particular applying the theorems of spectral graph theory. Conversely, for symmetric generating sets, the spectral and representation theory of Γ (G, S) {\displaystyle \Gamma (G,S)} are directly tied together: take ρ 1, …, ρ k {\displaystyle \rho _{1},\dots,\rho _{k}} a complete set of irreducible representations of G, {\displaystyle G,} and let ρ i (S) = ∑ s ∈ S ρ i (s) {\textstyle \rho _{i}(S)=\sum _{s\in S}\rho _{i}(s)} with eigenvalues Λ i (S) {\displaystyle \Lambda _{i}(S)}. Then the set of eigenvalues of Γ (G, S) {\displaystyle \Gamma (G,S)} is exactly ⋃ i Λ i (S), {\textstyle \bigcup _{i}\Lambda _{i}(S),} where eigenvalue λ {\displaystyle \lambda } appears with multiplicity dim (ρ i) {\displaystyle \dim(\rho _{i})} for each occurrence of λ {\displaystyle \lambda } as an eigenvalue of ρ i (S). {\displaystyle \rho _{i}(S).}

The genus of a group is the minimum genus for any Cayley graph of that group.
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