Isospectral Graph Reductions and Improved Estimates of Matrices' Spectra

TL;DR

This paper introduces isospectral graph reduction as a method to produce smaller matrices that preserve the spectrum, enabling improved eigenvalue estimates through iterative reductions.

Contribution

It demonstrates that eigenvalue estimates improve with each reduction, allowing more accurate spectral approximations of large matrices.

Findings

Eigenvalue estimates improve with graph reduction

Repeated reductions lead to more accurate eigenvalue bounds

The method enhances spectral analysis of large matrices

Abstract

Via the process of isospectral graph reduction the adjacency matrix of a graph can be reduced to a smaller matrix while its spectrum is preserved up to some known set. It is then possible to estimate the spectrum of the original matrix by considering Gershgorin-type estimates associated with the reduced matrix. The main result of this paper is that eigenvalue estimates associated with Gershgorin, Brauer, Brualdi, and Varga improve as the matrix size is reduced. Moreover, given that such estimates improve with each successive reduction, it is also possible to estimate the eigenvalues of a matrix with increasing accuracy by repeated use of this process.