Spectra and eigenspaces from regular partitions of Cayley (di)graphs of permutation groups

Abstract In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on n letters. We prove that every partition of the number n gives rise to a regular partition of the Cayley graph. By using representation theory, we also obtain the complete spectra and the eigenspaces of the corresponding quotient (di)graphs. More precisely, we provide a method to find all the eigenvalues and eigenvectors of such (di)graphs, based on their irreducible representations. As examples, we apply this method to the pancake graphs P ( n ) and to a recent known family of mixed graphs Γ ( d , n , r ) (having edges with and without direction). As a byproduct, the existence of perfect codes in P ( n ) allows us to give a lower bound for the multiplicity of its eigenvalue −1.