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 set of vertices and E(G) is a set of edges called a graph. We denote a vertex by v, where v 2 V(G) and edge by e, where e = uv 2 E(G). We denote degree of vertex v by dv which is defined as the number of edges adjacent with vertex v. The distance between two vertices of G is the length of a shortest path connecting these two vertices which is denoted by d(u,v) where u,v 2 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 nanostar dendrimers and then we compute its second eccentric Zagreb index. Ghorbani and Hosseinzadeh introduced the second eccentric Zagreb index as EM2(G) = åuv2E(G) (ecc(u) ecc(v)),that ecc(u) denotes the eccentricity of vertex u and ecc(v) denotes the eccentricity of vertex v of G.