# Extensions and Applications of Equitable Decompositions for Graphs with Symmetries

## TL;DR

This paper extends equitable decomposition theory for graphs with symmetries, allowing eigenvalue and eigenvector analysis to be simplified through automorphisms, with applications to spectral radius and eigenvalue localization.

## Contribution

It introduces the concept of separable automorphisms, enabling equitable decomposition over complex automorphisms, and shows how this affects eigenvalues, eigenvectors, and Gershgorin regions.

## Key findings

- Eigenvalues are preserved under equitable decomposition.
- Eigenvectors can be decomposed alongside matrices.
- Gershgorin regions are contained within original regions, aiding eigenvalue localization.

## Abstract

We extend the theory of equitable decompositions, in which, if a graph has a particular type of symmetry, i.e. a uniform or basic automorphism $\phi$, it is possible to use $\phi$ to decompose a matrix $M$ appropriately associated with the graph. The result is a number of strictly smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix $M$. We show here that a large class of automorphisms, which we refer to as \emph{separable}, can be realized as a sequence of basic automorphisms, allowing us to equitably decompose $M$ over any such automorphism. We also show that not only can a matrix $M$ be decomposed but that the eigenvectors of $M$ can also be equitably decomposed. Additionally, we prove under mild conditions that if a matrix $M$ is equitably decomposed the resulting divisor matrix, which is the divisor matrix of the associated equitable partition, will have the same spectral radius as the original matrix $M$. Last, we describe how an equitable decomposition effects the Gershgorin region $\Gamma(M)$ of a matrix $M$, which can be used to localize the eigenvalues of $M$. We show that the Gershgorin region of an equitable decomposition of $M$ is contained in the Gershgorin region $\Gamma(M)$ of the original matrix. We demonstrate on a real-world network that by a sequence of equitable decompositions it is possible to significantly reduce the size of a matrix' Gershgorin region.