THE HOSOYA POLYNOMIAL OF TUC4C8(S) NANOTUBES

One of the main distinctive characteristics of modern chemistry is the use of theoretical tools for the molecular modeling of physicochemical processes, chemical reaction, medicinal and toxicological events, etc., in which chemicals are involved. Topological indices are one of the main theoretical tools for studying molecular properties of chemical compounds. Here, a topological index is a real number that is derived from molecular graphs of chemical compounds. Such numbers based on the distances in a graph are widely used for establishing relationships between the structure of molecules and their physico-chemical properties. It is easy to see that the number of atoms and the number of bonds in a molecular graph are topological index. The first non trivial topological index was introduced early by Wiener. He defined his index as the sum of distances between any two carbon atoms in the molecules, in terms of carbon-carbon bonds. We encourage the reader to consult papers and references therein, for further study on the topic. Let G be a simple molecular graph without directed and multiple edges and without loops, the vertex and edge sets of which are represented by V(G) and E(G), respectively. If e is an edge of G, connecting the vertices u and v then we write e = uv. The distance between a pair of vertices u and v of G is denoted by d(u,w). Thus, we can redefine the Wiener index of a graph G as W(G) = ∑{x,y}⊆V(G)d(x,y). The Hosoya polynomial of a molecular graph G is defined as where the sum is over all unordered pairs {u,v} of distinct vertices in G. Suppose D = [dij] denotes the distance matrix of G, where dij is the length of a minimal path connecting the ith and jth vertices of G. Then one can see that W(G) = 1/2∑i,jdij and H(G,x) = 1/2∑i,j . , x x) H(G, ) G ( V } v , u { v) d(u, ∑ ⊆ =