Calculation of the Wiener, Szeged, and $PI$ Indices of a Certain Nanostar Dendrimer

Abstract

Let $G=(V,E)$ be a simple connected graph with vertex set $V$ and edge set $E$. The Wiener index of $G$ is defined by $W(G)=\sum _{x,y\subseteq V}d(x,y),$ where $d(x,y)$ is the length of the shortest path from $x$ to $y$. The Szeged index of $G$ is defined by $S_z(G)=\sum _{e=uv\in E}{n}_u(e|G)n_v(e|G),$ where $n_u(e|G)$ (resp. $n_v(e|G)$) is the number of vertices of $G$ closer to $u$ (resp. $v$) than $v$ (resp. $u$). The Padmakar-Ivan index of $G$ is defined by $PI(G)=\sum _{e=uv\in E}[n_{eu}(e|G)+n_{ev}(e|G)],$ where $n_{eu}(e|G)$ (resp. $n_{ev}(e|G)$) is the number of edges of $G$ closer to $u$ (resp. $v$) than $v$ (resp. $u$). In this paper, we will consider the graph of a certain nanostar dendrimer consisting of a chain of hexagons and find its topological indices such as the Wiener, Szeged, and $PI$ index.