Spectral properties of the exponential distance matrix

TL;DR

This paper investigates the spectral properties of the exponential distance matrix in graphs, establishing characteristic polynomial properties, identifying uniquely determined graphs, and constructing cospectral examples.

Contribution

It provides new insights into the spectrum of the exponential distance matrix, including spectral characterization and cospectral graph constructions.

Findings

Characterizes the spectrum of the exponential distance matrix

Identifies graph families uniquely determined by their spectrum

Constructs examples of cospectral graphs

Abstract

Given a graph $G$, the exponential distance matrix is defined entry-wise by letting the $(u,v)$-entry be $q^{\text{dist}(u,v)}$, where $\text{dist}(u,v)$ is the distance between the vertices $u$ and $v$ with the convention that if vertices are in different components, then $q^{\text{dist}(u,v)}=0$. In this paper, we will establish several properties of the characteristic polynomial (spectrum) for this matrix, give some families of graphs which are uniquely determined by their spectrum, and produce cospectral constructions.