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 \Gamma(G,S) are directly tied together: take \rho_1,\dots,\rho_k a complete set of irreducible representations of G, and let \rho_i(S) = \sum_{s \in S} \rho_i(s) with eigenvalues \Lambda_i(S). Then the set of eigenvalues of \Gamma(G,S) is exactly \bigcup_i \Lambda_i(S), where eigenvalue \lambda appears with multiplicity \dim(\rho_i) for each occurrence of \lambda as an eigenvalue of \rho_i(S).
