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.