Eigenvectors of isospectral graph transformations

TL;DR

This paper explores the relationship between eigenvectors of original graphs and their isospectral reductions, providing insights and an application for updating maximal eigenvectors in large sparse networks.

Contribution

It introduces a simple observation linking eigenvectors before and after isospectral graph reduction, enhancing the existing theory and enabling efficient eigenvector updates.

Findings

Established a relation between eigenvectors of original and reduced graphs

Proposed an algorithm for updating maximal eigenvectors in large networks

Enhanced understanding of isospectral graph transformations

Abstract

L.A. Bunimovich and B.Z. Webb developed a theory for isospectral graph reduction. We make a simple observation regarding the relation between eigenvectors of the original graph and its reduction, that sheds new light on this theory. As an application we propose an updating algorithm for the maximal eigenvector of the Markov matrix associated to a large sparse dynamical network.