Proximity and Remoteness in Directed and Undirected Graphs

TL;DR

This paper investigates the extremal properties of average distances in strongly connected directed and undirected graphs, establishing bounds, characterizing extremal structures, and exploring conditions for equality of proximity and remoteness.

Contribution

It provides sharp bounds on proximity and remoteness for strongly connected digraphs and tournaments, characterizes when these measures are equal, and introduces non-regular examples with equal proximity and remoteness.

Findings

Sharp bounds on proximity and remoteness as functions of graph order.

Characterization of when a strong tournament has equal proximity and remoteness.

Existence of non-regular strong digraphs with equal proximity and remoteness.

Abstract

Let $D$ be a strongly connected digraph. The average distance $\bar{\sigma}(v)$ of a vertex $v$ of $D$ is the arithmetic mean of the distances from $v$ to all other vertices of $D$. The remoteness $\rho(D)$ and proximity $\pi(D)$ of $D$ are the maximum and the minimum of the average distances of the vertices of $D$, respectively. We obtain sharp upper and lower bounds on $\pi(D)$ and $\rho(D)$ as a function of the order $n$ of $D$ and describe the extreme digraphs for all the bounds. We also obtain such bounds for strong tournaments. We show that for a strong tournament $T$, we have $\pi(T)=\rho(T)$ if and only if $T$ is regular. Due to this result, one may conjecture that every strong digraph $D$ with $\pi(D)=\rho(D)$ is regular. We present an infinite family of non-regular strong digraphs $D$ such that $\pi(D)=\rho(D).$ We describe such a family for undirected graphs as well.