A relation between proximity and the third largest distance eigenvalue of a graph

Abstract

Proximity π and remoteness ρ are respectively the minimum and the maximum, over the vertices of a connected graph, of the average distance from a vertex to all others. The distance eigenvalues of a connected graph G, denoted by ∂1≥∂2≥⋯≥∂n, are those of its distance matrix. In this paper, we prove π+∂3>0 for any graph with a diameter at least 3. This result leads to relations between ∂3 and several distance invariants of a graph such as remoteness, diameter, radius, average eccentricity and average distance. In particular, it confirms ρ+∂3>0 for any graph with a diameter at least 3 conjectured by Aouchiche and Hansen (2016).