General Equitable Decompositions for Graphs with Symmetries

TL;DR

This paper extends equitable decomposition theory to any automorphism of a graph, providing a general method to decompose associated matrices while preserving spectral properties, especially the spectral radius.

Contribution

It introduces a universal equitable decomposition method applicable to all graph automorphisms, broadening previous restrictions and including a step-by-step procedure.

Findings

Decomposition preserves eigenvalues of the original matrix.

The divisor matrix shares the same spectral radius as the original.

The method applies to any automorphism without restrictions.

Abstract

Using the theory of equitable decompositions it is possible to decompose a matrix $M$ appropriately associated with a given graph. The result is a collection of smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix $M$. This is done by decomposing the matrix over a graph symmetry. Previously it was shown that a matrix can be equitably decomposed over any uniform, basic, or separable automorphism. Here we extend this theory to show that it is possible to equitably decompose a matrix over any automorphism of a graph, without restriction. Moreover, we give a step-by-step procedure which can be used to generate such a decomposition. We also 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 $M$.