Remoteness, order, size and connectivity constraints in digraphs

Let $D$ be a strongly connected digraph. The average distance of a vertex $v$ in $D$ is defined as the arithmetic mean of the distances from $v$ to all other vertices in $D$. The remoteness $\rho (D)$ of $D$ is the maximum of the average distances of the vertices in $D$. In this paper, we provide a sharp upper bound on the remoteness of a strong digraph with given order, size, and vertex-connectivity. We then characterise the extremal digraphs that maximise remoteness among all strong digraphs of order $n$, size at least $m$, and vertex-connectivity $\kappa$. Finally, we demonstrate that the upper bounds on the remoteness of a graph given its order, size, and connectivity constraints (see \cite{DanMafMal2025}) can be extended to a larger class of digraphs containing all graphs, the Eulerian digraphs.