A construction of distance cospectral graphs

TL;DR

This paper presents methods to construct pairs of graphs that have identical distance spectra despite differences in their edge counts, expanding understanding of distance cospectral graphs.

Contribution

It introduces new constructions and a subgraph switching technique for generating distance cospectral graphs, based on perturbing eigenvectors.

Findings

Constructed graphs with different edge counts but identical distance spectra.

Identified a subgraph switching method for creating additional distance cospectral graphs.

Provided proofs using eigenvector perturbation techniques.

Abstract

The distance matrix of a connected graph is the symmetric matrix with columns and rows indexed by the vertices and entries that are the pairwise distances between the corresponding vertices. We give a construction for graphs which differ in their edge counts yet are cospectral with respect to the distance matrix. Further, we identify a subgraph switching behavior which constructs additional distance cospectral graphs. The proofs for both constructions rely on a perturbation of (most of) the distance eigenvectors of one graph to yield the distance eigenvectors of the other.