Eccentric Connectivity Index of Cartesian product and Strong product of Strongly Connected Digraphs

Let $G=(V,E)$ be a graph. The \emph{eccentric connectivity index} of $G$ is defined as $${\xi }^C(G)=\sum _{u\in V(G)}{d}_{u}ecc(u)$$ where $d_u$ and $ecc(u)$ are the degree and eccentricity of $u$, respectively. For a strongly connected digraph $D=(V,A)$, the eccentric connectivity index is defined as $${\xi }^C(D)=\frac{1}{2}\sum _{u\in V(D)}(d_u^{+}+d_u^{-})mecc(u)$$ where $d_u^{+}$ and $d_u^{-}$ are the out-degree and in-degree of $u$, respectively, and $mecc(u)$ is its m-eccentricity with respect to the maximum distance $md(u,v)=\max \{\vec{d}(u,v),\vec{d}(v,u)\}$. In this article, give formulas and bounds for the eccentric connectivity index of Cartesian and strong products of strongly connected digraphs and discuss the corresponding equality cases. Also, an attempt is made to study the self-centeredness of these products and establish conditions under which the Cartesian and strong products are self-centered.The results are extended to products of several digraphs.