Remoteness, proximity and few other distance invariants in graphs

TL;DR

This paper proves conjectures related to extremal properties of distance invariants in graphs, specifically identifying trees and graphs that maximize or minimize differences between various distance measures.

Contribution

It establishes maximal and minimal trees and graphs for differences of average distance, proximity, eccentricity, remoteness, and radius, confirming several conjectures for trees.

Findings

Maximal trees for the difference of average distance and proximity are identified.

Maximal trees for the difference of average eccentricity and remoteness are established.

Minimal trees for the difference of remoteness and radius are proved.

Abstract

We establish maximal trees and graphs for the difference of average distance and proximity proving thus the corresponding conjecture posed in [4]. We also establish maximal trees for the difference of average eccentricity and remoteness and minimal trees for the difference of remoteness and radius proving thus that the corresponding conjectures posed in [4] hold for trees.