发表机构
NSS College Ottappalam; Department of Mathematics, Government College Chittur; University of Calicut; Department of Collegiate Education, Government of Kerala（N.S.S. 奥特帕拉姆学院; 奇图尔政府学院数学系; 卡利卡特大学; 喀拉拉邦 collegiate 教育部）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究强连通有向图的笛卡尔积和强积的偏心连通指数，给出其公式、界及等式条件，并探讨自中心性，推广至多个有向图的积。
AI 中文摘要
设$G=(V,E)$为一个图。$G$的偏心连通指数定义为$$\xi^C(G)=\sum_{u\in V(G)}d_uecc(u)$$其中$d_u$和$ecc(u)$分别是$u$的度和离心率。对于强连通有向图$D=(V,A)$，偏心连通指数定义为$$\xi^C(D)=\frac{1}{2}\sum_{u\in V(D)}(d_u^++d_u^-)mecc(u)$$其中$d_u^+$和$d_u^-$分别是$u$的出度和入度，$mecc(u)$是其关于最大距离$md(u,v)=\max\{\vec d(u,v),\vec d(v,u)\}$的m-离心率。本文给出了强连通有向图的笛卡尔积和强积的偏心连通指数的公式和界，并讨论了相应的等式成立情形。此外，还尝试研究这些积的自中心性，并建立了笛卡尔积和强积为自中心的条件。这些结果被推广到多个有向图的积。
英文摘要
Let $G=(V,E)$ be a graph. The \emph{eccentric connectivity index} of $G$ is defined as $$ξ^C(G)=\sum_{u\in V(G)}d_uecc(u)$$ 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 $$ξ^C(D)=\frac{1}{2}\sum_{u\in V(D)}(d_u^++d_u^-)mecc(u)$$ 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.
