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 + ∂ ⌊ 7 d 8 ⌋ > 0 . In this paper, we confirm these two conjectures. Furthermore, we give lower bounds on ∂ n + ρ and ∂ 1 − ρ when G ≇ K n and the extremal graphs are characterized. [ABSTRACT FROM AUTHOR]