The generalized distance spectrum of a graph and applications

TL;DR

This paper introduces a unified framework for the generalized distance spectrum of graphs, providing explicit eigenvalue computations for many graph classes and demonstrating applications in ecology models and Markov Chain analysis.

Contribution

It generalizes existing spectral graph theory by defining the generalized distance spectrum and offers explicit eigenvalue formulas for broad graph classes.

Findings

Eigenvalues can be explicitly computed for many graphs.

Applications demonstrated in ecology competition models.

Useful for analyzing rapidly mixing Markov Chains.

Abstract

The generalized distance matrix of a graph is the matrix whose entries depend only on the pairwise distances between vertices, and the generalized distance spectrum is the set of eigenvalues of this matrix. This framework generalizes many of the commonly studied spectra of graphs. We show that for a large class of graphs these eigenvalues can be computed explicitly. We also present the applications of our results to competition models in ecology and rapidly mixing Markov Chains.