An asymptotic resolution of a problem of Plesn\'{i}k

Abstract

Fix $d \ge 3$. We show the existence of a constant $c>0$ such that any graph of diameter at most $d$ has average distance at most $d-c \frac{d^{3/2}}{\sqrt n}$, where $n$ is the number of vertices. Moreover, we exhibit graphs certifying sharpness of this bound up to the choice of $c$. This constitutes an asymptotic solution to a longstanding open problem of Plesn\'{i}k. Furthermore we solve the problem exactly for digraphs if the order is large compared with the diameter.