THE SZEGED AND WIENER NUMBERS OF WATER-SOLUBLE POLYARYL ETHER DENDRIMER NANOSTARS

Dendrimers are macromolecules comprised of a series of branches extending outward from an inner core. The word dendrimer originates from the Greek dendron, meaning “tree”. These mkolecules1 have attracted much attention because of their various electrical and optical properties. Suppose G is a simple graph, a graph without multiple edges and loops. The set of vertices and edges of G are denoted by V(G) and E(G), respectively. A topological index is a numeric quantity derived from the structural graph of a molecule. The number of vertices and edges are the simplest topological indices of graphs. The distance dG(u,v) (d(u,v) for short) between two vertices u, v ∈ V(G) is the length of a shortest path connecting them. The concept of “topological index” was first proposed by Haro Hosoya for characterizing the topological nature of a graph. Such graph invariants are usually related to the distance function and so named distance based topological index. The first topological index of this type was proposed in 1947 by the chemist Harold Wiener. It is defined as the sum of all distances between vertices of the graph under consideration. The Szeged index is a topological index introduced by Ivan Gutman. To define the Szeged index of a graph G, we assume that e = uv is an edge connecting the vertices u and v. Suppose nu(e) is the number of vertices of G lying closer to u than v and nv(e) is the number of vertices of G lying closer to v than u. Then the Szeged index of the graph G is defined as Sz(G) = ∑e=uv∈E(G)[nu(e)nv(e)]. Notice that vertices equidistant from u and v are not taken into account. The aim of this paper is to compute the Wiener and Szeged indices of a water-soluble polyaryl ether dendrimer G[n], see Figure 1. We encourage the reader to consult papers published bi Diudea and co-authors and our earlier papers for background material as well as basic computational techniques. Our notations are standard and taken mainly from the standard book of graph theory.