Iranian Journal of Mathematical Chemistry
Volume:3 Issue: 3, Autumn 2012

Let G be a graph. The first Zagreb M1(G) of graph G is defined as: M1(G) = uV(G) deg(u)2. In this paper, we prove that each even number except 4 and 8 is a first Zagreb index of a caterpillar. Also, we show that the fist Zagreb index cannot be an odd number. Moreover, we obtain the fist Zagreb index of some graph operations.

The eccentricity connectivity index of a molecular graph G is defined as (G) = a (G) deg(a)ε(a), where ε(a) is defined as the length of a maximal path connecting a to other vertices of G and deg(a) is degree of vertex a. Here, we compute this topological index for some infinite classes of dendrimer graphs.

It is necessary to generate the automorphism group of a chemical graph in computer aided structure elucidation. An Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix M = [dij], where for i≠j, dij is the Euclidean distance between the nuclei i and j. In this matrix dii can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for distinct nuclei. A.T. Balaban introduced some monster graphs and then M. Randic computed complexity indices of them (see A.T. Balaban, Rev. Roum. Chim. 18(1973) 841-853 and M. Randic, Croat. Chem. Acta 74(3)(2001) 68 705). In this paper, we describe a simple method, by means of which it is possible to calculate the automorphism group of weighted graphs.

In this paper, some applications of our earlier results in working with chemical graphs are presented.

A graph that contains a Hamiltonian cycle is called a Hamiltonian graph. In this paper we compute the first and the second geometric – arithmetic indices of Hamiltonian graphs. Then we apply our results to obtain some bounds for fullerene.

The Omega polynomial(x) was recently proposed by Diudea, based on the length of strips in given graph G. The Sadhana polynomial has been defined to evaluate the Sadhana index of a molecular graph. The PI polynomial is another molecular descriptor. In this paper we compute these three polynomials for some infinite classes of nanostructures.

The topological index of a graph G is a numeric quantity related to G which is invariant under automorphisms of G. The Tutte polynomial of 􀜩 is a polynomial in two variables defined for every undirected graph contains information about connectivity of the graph. The Padmakar-Ivan, vertex Padmakar-Ivan polynomials of a graph 􀜩 are polynomials in one variable defined for every simple connected graphs that are undirected. In this paper, we compute these polynomials of two infinite classes of dendrimer nanostars.

The atom bond connectivity index of a graph is a new topological index was defined by E.Estrada as ABC(G) uvE (dG(u) dG(v) 2) / dG(u)dG(v), where G d (u) denotes degree of vertex u. In this paper we present some bounds of this new topological index.