Remoteness and distance eigenvalues of a graph

Abstract

Let $G$ be a connected graph of order $n$ with diameter $d$. Remoteness $\ddot{I}$ of $G$ is the maximum average distance from a vertex to all others and ${\partial }_1\ge \cdots \ge {\partial }_n$ are the distance eigenvalues of $G$. In \cite{AH}, Aouchiche and Hansen conjectured that $\ddot{I}+{\partial }_3>0$ when $d\ge 3$ and $\ddot{I}+{\partial }_{\lfloor \frac{7d}{8}\rfloor }>0.$ In this paper, we confirm these two conjectures. Furthermore, we give lower bounds on ${\partial }_n+\ddot{I}$ and ${\partial }_1-\ddot{I}$ when $G≆K_n$ and the extremal graphs are characterized.

9 pages