On the difference between proximity and other distance parameters in triangle-free graphs and $C_4$-free graphs

TL;DR

This paper investigates the relationships between proximity, remoteness, diameter, and radius in triangle-free and $C_4$-free graphs, providing sharp bounds and exploring how these parameters differ.

Contribution

It introduces new upper bounds on the differences between proximity and other distance parameters in specific graph classes, with proofs of sharpness.

Findings

Bounds on the difference between remoteness and proximity.

Bounds on the difference between diameter and proximity.

Bounds on the difference between radius and proximity.

Abstract

The average distance of a vertex $v$ of a connected graph $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity $\pi(G)$ and the remoteness $\rho(G)$ of $G$ are the minimum and the maximum of the average distances of the vertices of $G$. In this paper, we give upper bounds on the difference between the remoteness and proximity, the diameter and proximity, and the radius and proximity of a triangle-free graph with given order and minimum degree. We derive the latter two results by first proving lower bounds on the proximity in terms of order, minimum degree and either diameter or radius. Our bounds are sharp apart from an additive constant. We also obtain corresponding bounds for $C_4$-free graphs.