Proximity, remoteness and maximum degree in graphs

TL;DR

This paper establishes sharp upper bounds on the proximity and remoteness of connected graphs based on their order, minimum degree, and maximum degree, contributing to the understanding of graph distance properties.

Contribution

It provides new upper bounds on proximity and remoteness for graphs with specified parameters, improving theoretical understanding of graph distance measures.

Findings

Upper bounds on remoteness and proximity are sharp up to an additive constant.

Bounds depend on graph order, minimum degree, and maximum degree.

Results enhance the theoretical framework of graph distance metrics.

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 $G$, respectively. In this paper, we give upper bounds on the remoteness and proximity for graphs of given order, minimum degree and maximum degree. Our bounds are sharp apart from an additive constant.