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 Plesník. Furthermore we solve the problem exactly for digraphs if the order is large compared with the diameter.

中文翻译：

---

Plesník问题的渐近解决

固定 $d\ge 3$。我们展示了一个常数的存在$C>0$这样任何直径最大为d的图都具有最大平均距离$d-C\frac{d^{3/2}}{\sqrt{ñ}}$，其中n是顶点数。此外，我们展示的图表证明了这种锐度取决于c的选择。这构成了一个长期的Plesník开放问题的渐近解决方案。此外，如果阶数与直径相比大，我们将为有向图精确地解决该问题。