Remoteness and distance eigenvalues of a graph

Let G be a connected graph of order n with diameter d. Remoteness ρ of G is the maximum average distance from a vertex to all others and ∂1≥⋯≥∂n are the distance eigenvalues of G. Aouchiche and Hansen (0000), Aouchiche and Hansen conjectured that ρ+∂3>0 when d≥3 and ρ+∂⌊7d8⌋>0. In this paper, we confirm these two conjectures. Furthermore, we give lower bounds on ∂n+ρ and ∂1−ρ when G≇Kn and the extremal graphs are characterized.