Remoteness and distance eigenvalues of a graph

TL;DR

This paper confirms conjectures relating remoteness and distance eigenvalues of a graph, providing bounds and characterizations for extremal graphs, advancing understanding of spectral graph properties.

Contribution

It proves two conjectures connecting remoteness and distance eigenvalues, and characterizes extremal graphs with bounds on these spectral parameters.

Findings

Confirmed that + ho>0 for graphs with diameter  or more.

Established lower bounds on + ho and - ho for non-complete graphs.

Characterized extremal graphs achieving these bounds.

Abstract

Let $G$ be a connected graph of order $n$ with diameter $d$. Remoteness $\rho$ of $G$ is the maximum average distance from a vertex to all others and $\partial_1\geq\cdots\geq \partial_n$ are the distance eigenvalues of $G$. In \cite{AH}, Aouchiche and Hansen conjectured that $\rho+\partial_3>0$ when $d\geq 3$ and $\rho+\partial_{\lfloor\frac{7d}{8}\rfloor}>0.$ In this paper, we confirm these two conjectures. Furthermore, we give lower bounds on $\partial_n+\rho$ and $\partial_1-\rho$ when $G\ncong K_n$ and the extremal graphs are characterized.