Extremal total distance of graphs of given radius I

TL;DR

This paper determines the extremal total distance for graphs with a given order and radius, confirming a conjecture asymptotically and extending results to digraphs for large orders.

Contribution

It proves the asymptotic extremal total distance for graphs of given order and radius and extends the conjecture to digraphs.

Findings

Confirmed the conjecture for graphs with large order

Extended the extremal total distance results to digraphs

Provided characterizations of extremal graphs and digraphs

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 also state an analog of the conjecture of Chen et al for digraphs and prove it for sufficiently large order.