Computing Wiener and Schultz indices of HA[C.sub.5][C.sub.7] [p, q] nanotube by GAP program

INTRODUCTION

Topological indices of nanotubes are numerical descriptors that are derived from graph of chemical compounds. Such indices based on the distances in graph are widely used for establishing relationships between the structure of nanotubes and their physicochemical properties. Usage of topological indices in biology and chemistry began in 1947 when chemist Harold Wiener (1) introduced Wiener index to demonstrate correlations between physicochemical properties of organic compounds and the index of their molecular graphs. Wiener originally defined his index (W) on trees and studied its use for correlations of physicochemical properties of alkanes, alcohols, amines and their analogous compounds (2).

Let G be a connected graph. The vertex-set and edge-set of G denoted by V(G) and E(G), respectively. The degree of a vertex i [member of] V(G) is the number of vertices joining to i and denoted by v(i). The (i, j) entry of the adjacency matrix of G is denoted by A (I, j) The Wiener index of a graph G is denoted by W (G) and defined as the sum of distances between all pairs of vertices in G:

W(G) = 1/2 [summation over ((i,j)[]V(G))] d(i,j)(1)

where, d(i, j) is the distance between vertices i and j. Another topological index is Schultz index, the Schultz index (MTI) was introduced by Schultz in 1989, as the molecular topological index (3) and it is defined by:

MTI = [summation over ((i,j) V(G))] v(i)d(i,j) + A(i,j).(2)

The molecular topological index studied in many papers (4-7).

In a series of papers, the Wiener index of some nanotubes is computed (8-14), another topological indices are computed (15-19). In this research, we give an algorithm for computing the Wiener and Schultz indices of any graph and by this algorithm; we obtain the Wiener and Schultz indices of HA[C.sub.5][C.sub.7] [p, q] nanotube.

AN ALGORITHM FOR THE COMPUTATION OF THE WIENER AND SCHULTZ INDICES OF A GRAPH

Here, we give an algorithm that enables us to compute the Wiener and Schultz indices of any graph. For this purpose, the following algorithm is presented:

* We assign to any vertex one number

* We determine all of adjacent vertices set of the vertex i, i[member of] V and this set denoted by N(i)

* In the start of program, we set w = 0, Sc = 0 and at the end of program, the values of 1/2 w and Sc will be the Wiener and Schultz indices of graph G respectively

The set of vertices that their distance to vertex i is equal to t (t[greater than or equal to]0) is denoted by [D.sub.i,t] and consider [D.sub.i,o] = {i} . We have following relations:

V = [U.sub.t[greater than or equal to]0][D.sub.i,t]i[member of] V (3)

[summation over (j [member of]V(G))]d(i,j) = [summation over (t[greater than or equal to]1)]tX|[D.sub.i,t]|, [for all]i[member of]V(G) (4)

W(G) = [1/2][summation over (i[euro]V,t[greater than or equal to]1)]tX|[D.sub.i,t] (5)

MTI(G) = [summation over (i[member of]V(G))]v(i)X[summation over (j[member of]V(G))]d(i,j) + A(i,j)) = [summation over (i[member of]V(G))]v(i)X([summation over (j[member of]N(i))]2 + [summation over (j[member of]Y(G)\N(i))]d(i,j)) = [summation over (i[member of]V(G))](2v[(i).sup.2] + v(i)X[summation over (j[member of][D.sub.i - t]t[greater than or equal to]2)]tX|[D.sub.i,t]|) (6)

According to Eq. (5) and (6), by determining these sets, we can obtain the wiener and Schultz indices of the graph.

The distance between vertex i and its adjacent vertices is equal to 1, therefore [D.sub.i,1] = N(i). For each j [member of] [D.sub.i,t] t [greater than or equal to] 1, the distance between each vertex of set N( j)\([D.sub.i,t] [union] [D.sub.i, t-1]) and the vertex i is equal to t+1, thus we have

[D.sub.i,t + 1] = [U.j[member of] [D.sub.i,t],t[greater than or equal to]1](N(j)\([D.sub.i,t]U[D.sub.i,t - 1])) (7)

According to Eq. (7), we can obtain [D.sub.i,t], t [greater than or equal to] 2 for each i [member of] V. In this step, we compute the wiener and Schultz indices of the graph by above relations.

COMPUTING THE WIENER AND SCHULTZ INDICES OF HA[C.sub.5][C.sub.7] [P, Q] NANOTUBE

A [C.sub.5][C.sub.7] net is a trivalent decoration made by alternating [C.sub.5] and [C.sub.7]. It can cover either a cylinder or a torus. Here we compute the Wiener and Schultz indices of HA[C.sub.5][C.sub.7] [p, q] nanotube by GAP program (Fig. 1, Table 1).

We denote the number of heptagons in one row by p. In this nanotube, the three first rows of vertices and edges are repeated alternatively and we denote the number of this repetition by q. In each period there are 8p vertices and p vertices which are joined to the end of the graph and hence the number of vertices in this nanotube is equal to 8pq+p.

We partition the vertices of this graph to following sets:

[K.sub.1]: The vertices of first row whose number is 2p.

[K.sub.2]: The vertices of the first row in each period except the first one whose number is 2p(q-1).

[K.sub.3]: The vertices of the second rows in each period whose number is 3pq.

Table1: Wiener and Schultz indices of HA[C.sub.5][C.sub.7][p, q]
nanotube

p  q    W(G)     MTI(G)

3  1     1167       246
4  2    12236     68376
4  3    35052    199672
5  3    57915    330040
6  4   187068   1078044
7  4   265391   1530130
7  7  1220219   7143010
3  6   134787    786006
4  6   243276   1418392
4  7   379756   2222456
5  7   601855   3522320
6  8  1290348   7573884
7  8  1781535  10458098
7  9  2495731  14683410
8  8  2362824  13872176
9  9  4235085  24921882