Extremal total distance of graphs of given radius

TL;DR

This paper investigates the extremal total distance in graphs with a fixed radius, establishing asymptotic results and confirming a conjecture, while also linking total distance minimization to size maximization.

Contribution

It proves the asymptotic extremal total distance for graphs with given order and radius, confirming a conjecture and exploring related bounds.

Findings

Confirmed asymptotic extremal total distance for large graphs

Established connection between total distance minimization and size maximization

Provided bounds for maximum total distance

Abstract

In 1984, Plesn\'{i}k determined the minimum total distance for given order and diameter and characterized the extremal graphs and digraphs. We prove the analog for given order and radius, when the order is sufficiently large compared to the radius. This confirms asymptotically a conjecture of Chen et al. We show the connection between minimizing the total distance and maximizing the size under the same conditions. We also prove some asymptotically optimal bounds for the maximum total distance.