Let $D$ be a strongly connected digraph. The average distance $\bar{\sigma}(v)$ of a vertex $v$ of $D$ is the arithmetic mean of the distances from $v$ to all other vertices of $D$. The remoteness $\rho(D)$ and proximity $\pi(D)$ of $D$ are the maximum and the minimum of the average distances of the vertices of $D$, respectively. We obtain sharp upper and lower bounds on $\pi(D)$ and $\rho(D)$ as a function of the order $n$ of $D$ and describe the extreme digraphs for all the bounds. We also obtain such bounds for strong tournaments. We show that for a strong tournament $T$, we have $\pi(T)=\rho(T)$ if and only if $T$ is regular. Due to this result, one may conjecture that every strong digraph $D$ with $\pi(D)=\rho(D)$ is regular. We present an infinite family of non-regular strong digraphs $D$ such that $\pi(D)=\rho(D).$ We describe such a family for undirected graphs as well.

中文翻译：

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有向图和无向图中的邻近度和远程度

令 $D$ 是一个强连通有向图。$D$的顶点$v$的平均距离$\bar{\sigma}(v)$是$v$到$D$的所有其他顶点的距离的算术平均值。$D$的远距离$\rho(D)$和接近$\pi(D)$分别是$D$顶点平均距离的最大值和最小值。我们在 $\pi(D)$ 和 $\rho(D)$ 上获得了清晰的上限和下限，作为 $D$ 的 $n$ 阶的函数，并描述了所有边界的极端有向图。我们也为强大的比赛获得了这样的界限。我们证明，对于一个强大的锦标赛 $T$，我们有 $\pi(T)=\rho(T)$ 当且仅当 $T$ 是常规的。由于这一结果，人们可以推测每一个具有 $\pi(D)=\rho(D)$ 的强有向图 $D$ 都是正则的。我们提出了一个无限的非常规强有向图族 $D$，使得 $\pi(D)=\rho(D)。