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Proximity and Remoteness in Directed and Undirected Graphs

Let $D$ be a strongly connected digraph. The average distance $\barσ(v)$ of a vertex $v$ of $D$ is the arithmetic mean of the distances from $v$ to all other vertices of $D$. The remoteness $ρ(D)$ and proximity $π(D)$ 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 $π(D)$ and $ρ(D)$ 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 $π(T)=ρ(T)$ if and only if $T$ is regular. Due to this result, one may conjecture that every strong digraph $D$ with $π(D)=ρ(D)$ is regular. We present an infinite family of non-regular strong digraphs $D$ such that $π(D)=ρ(D).$ We describe such a family for undirected graphs as well.