Introduction

In this section we describe some notations which will be kept throughout. The icosahedral C₆₀ molecule discovered by Kroto et al. in 1985 [₁] was the first known example of a fullerene: a molecular cage of carbon consisting of fused hexagon and pentagon rings. The interest generated by this discovery led to the identification of many other fullerenes and related structures [₂], and in particular to the observation of carbon nanotubes. Fullerenes are a new allotrope of carbon characterized by a closed-cage structure consisting of an even number of three-coordinate carbon atoms devoid of hydrogen atoms. This class was originally limited to closed-cage structures with 12 isolated five-membered rings, the rest being six-membered rings. However, the term has been broadened to include any closed-cage structure having 20 or more carbon atoms consisting entirely of 3 coordinate carbon atoms [₃]. The motivation for the study of fullerenes is outlined in the nice book of Korto, Fichier and Cox [₄], and the reader is encouraged to consult this book for background material.

Fullerenes with a wide range of numbers of carbon atoms have been produced in experiment. Isomers with 60, 70, 76, 78 and 84 atoms have been produced in sufficient quantity to be characterized by NMR spectroscopy.

The adjacency matrix or Hamiltonian operator A = [A_ij] of a graph G with n vertices is the square n × n symmetric matrix which contains information about the internal connectivity of vertices in the graph. It is defined as

Figure 1

One can see easily the fact that all unitary matrices commuting with the adjacency matrix A of a molecular graph form a group H with this property that:

Figure 2

The group H is called the Hamiltonian group of the graph under consideration and its elements are defined as a generalized symmetry operator. In a real vector space, these matrices are orthogonal. It is well known that the symmetry operators in the point group of a molecule always commute with its Hamiltonian operator [₅]. So the group H of molecular graph must contain the point group H_p of the graph, for details see [₆]. It is well-known fact that the set of all permutation matrices satisfied in equation (1) constitute the full automorphism group of the molecular graph under consideration. Here a permutation matrix is a matrix that has exactly one 1 in each row or column and 0s elsewhere. Permutation matrices are the matrix representation of permutations.

A topological index is a real number related to a molecular graph. It must be a structural invariant, i.e., it does not depend on the labelling or the pictorial representation of a graph. There are several topological indices have been defined and many of them have found applications as means to model chemical, pharmaceutical and other properties of molecules. Here, we consider three of topological indices containing Wiener, Padmakar-Ivan (PI) and Schultz index. These topological indices define as follows:

The Wiener index W(G) of a molecular graph G is defined as the sum of the distances between all pairs of vertices. In other words,

Figure 3

where P_i is length of the path that contains the least number of edges between vertex i and vertex j in graph G and n is the maximum possible number of i and j, see [₇,₈,₉] for details.

The Padmakar–Ivan (PI) index of a graph G is defined as PI(G) = ∑_e∈E[n_eu(e∣G)+ n_ev(e∣G)], where n_eu(e∣G) is the number of edges of G lying closer to u than to v, n_ev(e∣G) is the number of edges of G lying closer to v than to u and summation goes over all edges of G, [₁₀,₁₁,₁₂,₁₃,₁₄].

The Schultz index of a graph G is denoted by MTI(G). It is defined as

Figure 4

where v_i is the degree of vertex i in G, A_ij is equal to unity if vertices i and j are adjacent and zero otherwise and D_ij is the length of a shortest path between vertices i and j, [₁₅,₁₆].

The goal of this article is to obtain some computer programs for calculating the Wiener, PI and Schultz indices of molecular graphs. We apply this program on some big fullerenes. We also apply our earlier MATLAB Program [₁₇,₁₈] for computing the automorphism group of fullerenes C₂₀(1), C₂₀(2), C₆₀, C₇₀, C₈₀, C₁₄₀ and C₁₅₀. Throughout this paper, only finite graphs are investigated. Our notation is standard and taken mainly from Refs. [₁₉,₂₀]. Computations were carried out with the aid of MATLAB [₂₁], and, GAP [₂₂]. We encourage reader to consult [₂₃,₂₄] for applications of GAP in solving problems of chemistry and [₂₅,₂₆,₂₇,₂₈,₂₉] for background material about theoretical aspects of symmetry property of molecules.

Computational Details

Permutation group theory has been widely applied to various chemical problems related to the symmetry property of a molecule. Symmetry operations on a graph are called also graph automorphisms. They affect only the labels of vertices by permuting them so that the adjacency matrix of the graph remains unchanged. The graph symmetry is completely determined by all the automorphisms it has, i.e. by specifying all the permutations which leave the adjacency matrix intact.

There are potential methods for computing automorphism group of a graph. We use these techniques to solve our problem. In order to deduce the symmetry of a molecule, we have to compute first the n × n adjacency matrix A of the chemical graph under consideration. Then we must solve the matrix equation P^tAP = P, in the set of all permutation matrices of size n.

The first author in [₁₇] proved a result that is useful for computing symmetry of molecules. Using this result, Lemma 1 and its Corollary, we present a MATLAB program [₁₈] for computing a solution matrix for the automorphism group of chemical graphs. We believe that MATLAB is the best software for writing such a program, because it uses variables that are defined to be matrices. Since the image of any vertex v of a graph under one of its automorphisms has the same degree as v, we can improve our last program in [₁₈] to find a better running time. In the end of this section we present our new MATLAB program, as follows:

Program: A MATLAB Program for Computing the Symmetries of Molecules

Symmetry and Topological Indices of Fullerenes

The aim of this section is to calculate the automorphism group of chemical graph of the fullerene C₂₀(1), C₂₀(2), C₆₀, C₇₀, C₈₀, C₁₄₀ and C₁₅₀. To do this, from figure 1, we calculate the adjacency matrices of these fullerene graphs and then apply our MATLAB program for computing a solution matrix for any of these fullerenes. In the final step, we consider the following simple GAP program for computing the automorphism group of these fullerenes.

Figure 5

Figure 1: Schlegel Diagrams of some Fullerenes

In this program, X denotes the solution matrix computed bt our MATLAB programs. We encourage the reader to consult Refs. [₂₃,₂₄] for more details about GAP and its programming language. In what follows, we calculate a generating set for any of automorphism groups of the fullerene graphs under consideration.

Generators of The Automorphism Groups of Fullerenes

C(1), C(2), C, C, C, C and C

A MATLAB Program for Computing the Wiener, PI and Schultz Indices

In Table 1, we calculate these topological indices, their running times and the order of symmetry group for the mentioned fullerenes.

Figure 6

Table 1: The Values of Wiener, PI and Schultz Indices of Fullerenes

C(1), C(2), C, C, C, C and C and the Order of Their Automorphism Group Symmetry

In the end of this paper we pose the following open questions:

Conjecture 1: If G is a fullerene graph then PI(G) < W(G), where W(G) denotes the Wiener index of G.

Conjecture 2: Suppose G₁, G₂, ... is a sequence of fullerene graphs such that all of them has a given group E as automorphism group and ∣V(G₁)∣, ∣V(G₂)∣, ... is an increasing sequence of natural numbers. Then

Figure 7

We mention here that the figure of fullerenes in Figure 1 taken from the internet. Unfortunately, we don't have its exact address to cite here.