Zagreb Eccentricity Indices of the Generalized Hierarchical Product Graphs and Their Applications

Abstract

Let $G$ be a connected graph. The first and second Zagreb eccentricity indices of $G$ are defined as $M_{\mathrm{1}}^{\mathrm{*}}(G)={\sum }_{v\in V(G)}\mathrm{‍}{\epsilon }_G^{\mathrm{2}}(v)$ and $M_{\mathrm{2}}^{\mathrm{*}}(G)={\sum }_{uv\in E(G)}\mathrm{‍}{\epsilon }_G(u){\epsilon }_G(v)$, where ${\epsilon }_G(v)$ is the eccentricity of the vertex $v$ in $G$ and ${\epsilon }_G^{\mathrm{2}}(v)=({\epsilon }_G(v){)}^{\mathrm{2}}$. Suppose that $G(U)\sqcap H(\mathrm{\varnothing }\ne U\subseteq V(G))$ is the generalized hierarchical product of two connected graphs $G$ and $H$. In this paper, the Zagreb eccentricity indices $M_{\mathrm{1}}^{\mathrm{*}}$ and $M_{\mathrm{2}}^{\mathrm{*}}$ of $G(U)\sqcap H$ are computed. Moreover, we present explicit formulas for the $M_{\mathrm{1}}^{\mathrm{*}}$ and $M_{\mathrm{2}}^{\mathrm{*}}$ of S-sum graph, Cartesian, cluster, and corona product graphs by means of some invariants of the factors.