Remoteness and distance eigenvalues of a graph

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.