# A general method to obtain the spectrum and local spectra of a graph from its regular partitions

## TL;DR

This paper introduces a comprehensive method to derive the entire spectrum and local spectra of a graph using quotient matrices from regular partitions, applicable to various classes like walk-regular and distance-regular graphs.

## Contribution

It presents a new general approach to obtain all spectral information of a graph from its regular partitions, extending previous partial results.

## Key findings

- Method successfully computes full spectra of various graph classes.
- Applicable to walk-regular, distance-regular, and distance-biregular graphs.
- Provides explicit eigenvalues and multiplicities from quotient matrices.

## Abstract

It is well known that, in general, part of the spectrum of a graph can be obtained from the adjacency matrix of its quotient graph given by a regular partition. In this paper, we propose a method to obtain all the spectrum, and also the local spectra, of a graph $\Gamma$ from the quotient matrices of some of its regular partitions. As examples, it is shown how to find the eigenvalues and (local) multiplicities of walk-regular, distance-regular, and distance-biregular graphs.