THE SECOND ECCENTRIC ZAGREB INDEX OF THE $N^{TH}$ GROWTH OF NANOSTAR DENDRIMER $D_{3}[N]$

Abstract

Let $G=(V,E)$ be an ordered pair, where $V(G)$ is a non-empty setof vertices and $E(G)$ is a set of edges called a graph. We denotea vertex by $v$ where $v \in V(G)$ and edge by $e$ where $e=uv \inE(G)$. we denote degree of vertex $v$ by $d_{v}$ which is definedas the number of edges adjacent with vertex $v$. The distancebetween two vertices of $G$ is the length of a shortest pathconnecting these two vertices which is denoted by $d(u,v)$ where$u,v \in V(G)$. The eccentricity $ecc(v)$ of a vertex $v$ in $G$is the distance between vertex $v$ and vertex farthest from $v$ in$G$. In this paper, we consider an infinite family of NanostarDendrimers and compute its Second Eccentric Zagreb index.M.Ghorbani and Hosseinzadeh introduced Second eccentric zagrebindex as $EM_{2}(G)=\sum_{uv \in E(G)}\big(ecc(u)\timesecc(v)\big)$, that $ecc(u)$ denotes the eccentricity of a vertex$u$ and $ecc(v)$ denotes the eccentricity of a vertex $v$ of $G$.