Extremal total distance of graphs of given radius I

Abstract

In 1984, Plesník 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.

18 pages, 5 figures, this paper is an extended version of a first part of arXiv:1903.01358, mainly clarifying the content of sections 3 and 4 Figure 2 has been corrected, an Open access statement and a missing part of the introduction have been added