Eccentric Connectivity Index of Strongly Connected Digraphs

Let $G = (V, E)$ be a graph with non-empty set of vertices $V$ and set of edges $E$. The \emph{eccentric connectivity index} of the graph $G$ is defined as $$\displaystyle{\xi^C(G) = \sum_{u \in V} d_u \;ecc(u)}$$ where $d_u$ is the degree and $ecc(u)$ is the eccentricity of the vertex $u \in V$. This article is an attempt to find the \emph{eccentric connectivity index} of strongly connected digraph $D$ with respect to the metric, \textit{maximum distance} defined by $md(u,v)=\max\{\vec{d}(u,v),\vec{d}(v,u)\}$. An attempt is also made to find the extremal values for strongly connected digraphs.