On the Distance Spectra of Extended Double Stars

TL;DR

This paper proves that extended double star graphs are uniquely identified by their distance spectra, contributing to the understanding of which graphs are determined by their spectral properties.

Contribution

The paper establishes that extended double stars are uniquely determined by their distance spectra, advancing spectral graph theory.

Findings

Extended double stars are determined by their distance spectra.

The result helps classify graphs based on spectral properties.

It addresses a longstanding open problem in spectral graph theory.

Abstract

The distance matrix of a connected graph is defined as the matrix in which the entries are the pairwise distances between vertices. The distance spectrum of a graph is the set of eigenvalues of its distance matrix. A graph is said to be determined by its distance spectrum if there does not exist a non-isomorphic graph with the same spectrum. The question of which graphs are determined by their spectrum has been raised in the past, but it remains largely unresolved. In this paper, we prove that extended double stars are determined by their distance spectra.