Isospectral Graph Reductions

TL;DR

This paper introduces a flexible method for reducing complex weighted directed graphs to smaller ones while preserving their spectra, aiding in network design with specific spectral and dynamical features.

Contribution

It presents a novel general procedure for isospectral reduction of weighted digraphs, enabling spectrum preservation during graph simplification.

Findings

Reduces graph size without spectrum loss

Applicable to complex weighted digraphs

Facilitates network design with spectral constraints

Abstract

Let G be an arbitrary finite weighted digraph with weights in the set of complex rational functions. A general procedure is proposed which allows for the reduction of G to a smaller graph with a less complicated structure having the same spectrum as of G (up to some set known in advance). The proposed procedure has a lot of flexibility and could be used e.g. for design of networks with prescribed spectral and dynamical properties.