On the distance spectra of graphs

TL;DR

This paper studies the eigenvalues of the distance matrices of various classes of graphs, providing complete spectra for some distance-regular graphs and characterizing properties of strongly regular graphs.

Contribution

It determines the distance spectra of specific graph families and characterizes strongly regular graphs with certain eigenvalue properties, advancing spectral graph theory.

Findings

Distance spectra of halved cubes, double odd graphs, and Doob graphs are determined.

Characterization of strongly regular graphs with more positive than negative distance eigenvalues.

Determinant and inertia of distance matrices for lollipop and barbell graphs are computed.

Abstract

The distance matrix of a graph $G$ is the matrix containing the pairwise distances between vertices. The distance eigenvalues of $G$ are the eigenvalues of its distance matrix and they form the distance spectrum of $G$. We determine the distance spectra of halved cubes, double odd graphs, and Doob graphs, completing the determination of distance spectra of distance regular graphs having exactly one positive distance eigenvalue. We characterize strongly regular graphs having more positive than negative distance eigenvalues. We give examples of graphs with few distinct distance eigenvalues but lacking regularity properties. We also determine the determinant and inertia of the distance matrices of lollipop and barbell graphs.