Isospectral Reduction in Infinite Graphs

TL;DR

This paper extends isospectral reduction theory from finite graphs to certain infinite graphs and operators, enabling the calculation of stationary measures for countable Markov chains.

Contribution

It generalizes isospectral reduction to infinite graphs and Banach space operators, providing a new method to analyze infinite Markov chains.

Findings

Extended isospectral reduction to infinite graphs and operators

Calculated stationary measures for countable Markov chains

Demonstrated applicability to Banach space operators

Abstract

L. A. Bunimovich and B. Z. Webb developed a theory for transforming a finite weighted graph while preserving its spectrum, referred as isospectral reduction theory. In this work we extend this theory to a class of operators on Banach spaces that include Markov type operators. We apply this theory to infinite countable weighted graphs admitting a finite structural set to calculate the stationary measures of a family of countable Markov chains.