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

## TL;DR

This paper introduces a method to derive regular partitions, spectra, and eigenspaces of Cayley (di)graphs of permutation groups using representation theory, with applications to pancake graphs and mixed graphs.

## Contribution

It provides a novel approach to compute spectra and eigenspaces of Cayley (di)graphs based on regular partitions and irreducible representations, extending spectral analysis techniques.

## Key findings

- Complete spectra and eigenspaces of Cayley (di)graphs are obtained.
- Regular partitions correspond to partitions of the number n.
- Lower bounds for eigenvalue multiplicities are established.

## 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 $\Gamma(d,n,r)$ (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 $-1$.