A Relation Between Proximity And The Third Largest Distance Eigenvalue Of A Graph

Proximity pi and remoteness rho 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 partial derivative(1 )>= partial derivative(2) >=. . .>=partial derivative(n), are those of its distance matrix. In this paper, we prove pi + partial derivative(3) > 0 for any graph with a diameter at least 3. This result leads to relations between partial derivative(3) and several distance invariants of a graph such as remoteness, diameter, radius, average eccentricity and average distance. In particular, it confirms rho+partial derivative(3) > 0 for any graph with a diameter at least 3 conjectured by Aouchiche and Hansen (2016). (C) 2021 Elsevier B.V. All rights reserved.