Remoteness, order, size and connectivity constraints in digraphs

Abstract

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.