Iranian Journal of Mathematical Chemistry
Volume:1 Issue: 2, Autumn-Winter 2010

This note introduces a new general conjecture correlating the dimensionality dT of an infinite lattice with N nodes to the asymptotic value of its Wiener Index W(N). In the limit of large N the general asymptotic behavior W(N)≈Ns is proposed, where the exponent s and dT are related by the conjectured formula s=2+1/dT allowing a new definition of dimensionality dW=(s-2)-1. Being related to the topological Wiener index, dW is therefore called Wiener dimensionality. Successful applications of this method to various infinite lattices (like graphene, nanocones, Sierpinski fractal triangle and carpet) testify the validity of the conjecture for infinite lattices.

The concept of geometric−arithmetic indices (GA) was put forward in chemical graph theory very recently. In spite of this, several works have already appeared dealing with these indices. In this paper we present lower and upper bounds on the second geometric−arithmetic index (GA2) and characterize the extremal graphs. Moreover, we establish Nordhaus−Gaddum−type results for GA2.

Continuing the work K. C. Das, I. Gutman, B. Furtula, On second geometric−arithmetic index of graphs, Iran. J. Math Chem., 1 (2010) 17−27, in this paper we present lower and upper bounds on the third geometric−arithmetic index GA3 and characterize the extremal graphs.Moreover, we give Nordhaus−Gaddum−type result for GA3.

The Wiener index is a graph invariant that has found extensive application in chemistry. In addition to that a generating function, which was called the Wiener polynomial, who’s derivate is a q-analog of the Wiener index was defined. In an article, Sagan, Yeh and Zhang in [The Wiener Polynomial of a graph, Int. J. Quantun Chem., 60 (1996), 959−969] attained what graph operations do to the Wiener polynomial. By considering all the results that Sagan et al. admitted for Wiener polynomial on graph operations for each two connected and nontrivial graphs, in this article we focus on deriving Wiener polynomial of graph operations, Join, Cartesian product, Composition, Disjunction and Symmetric difference on n graphs and Wiener indices of them.

Eccentric connectivity index has been found to have a low degeneracy and hence a significant potential of predicting biological activity of certain classes of chemical compounds. We present here explicit formulas for eccentric connectivity index of various families of graphs.We also show that the eccentric connectivity index grows at most polynomially with thenumber of vertices and determine the leading coefficient in the asymptotic behavior.

Wiener index is a topological index based on distance between every pair of vertices in agraph G. It was introduced in 1947 by one of the pioneer of this area e.g, Harold Wiener. In the present paper, by using a new method introduced by Klavžar, we compute the Wiener, Szeged, GA, ABS, connectivity and Zagreb group indices of some nanostar dendrimers.

For a graph G with n vertices, its Estrada index is defined as EE(G) = Σ = ni e i 1 λ where n λ λ, λ, ..., 1 2 are the eigenvalues of G. A lot of properties especially lower and upper bounds for the Estrada index are known. We now establish further lower bounds for the Estrada index.

We derived explicit formulae for the eccentric connectivity index and Wiener index of2−dimensional square-octagonal TUC4C8(R) lattices with open and closed ends. Newcompression factors for both indices are also computed in the limit N→∞.

A novel algorithm for the fast detection of hubs in chemical networks is presented. Thealgorithm identifies a set of nodes in the network as most significant, aimed to be the most effective points of distribution for fast, widespread coverage throughout the system. We show that our hubs have in general greater closeness centrality and betweenness centrality than vertices with maximal degree, while having comparable or higher degree than vertices with greatest closeness centrality and betweenness centrality. As such, they serve as all purpose network hubs. Several theoretical and real world chemical and biological networks are tested and results are analyzed.

The general sum-connectivity index of a graph G, denoted by χ a = χ a (G) is definedas Σ ∈ = + () () uv E G a χ a du dv, where du (or dv) is the degree of the vertex u (or v). Efficient formulas for calculating the general sum-connectivity index of benzenoid systems and their phenylenes are given, and a relation is established between the general sum-connectivity index of a phenylene and of the corresponding hexagonal squeeze in this paper.

A topological index of a molecular graph G is a numeric quantity related to G which isinvariant under symmetry properties of G. Let G be a molecular graph. The eccentricconnectivity index ξ(G) is defined as () deg() (), () Σ ∈ ξ = ε u V G G u u where deg(u) denotes the degree of vertex u in G and ε(u) is the largest distance between u and any other vertex v of G. The augmented eccentric connectivity index, A ξ(G) is defined as Σ ∈ ε ξ = () () () G () u V A u G M u, where M(u) denotes the product of degrees of all neighbors of vertex u. In this paper, exact formulas for the eccentric connectivity and augmented eccentric connectivity indices of an infinite family of nanostar dendrimer are computed.

Topological indices are numerical parameters of a graph which characterize its topology. Inthis paper the PI, Szeged and Zagreb group indices of the tetrameric 1,3-adamantane arecomputed.