The multiplicative version of the Wiener index

The classical Wiener index, W(G), is equal to the sum of the distances between all pairs of vertexes of a (molecular) graph, G. We now consider a related topological index, pi(G), equal to the product of distances between all pairs of vertexes of G. The basic properties of the pi index are established and its possible physicochemical applications examined. In the case of alkanes, pi and W are highly correlated; a slightly curvilinear correlation exists between In pi and W.     

Modification of the Wiener index 4

A novel topological index W(F) is defined by the matrices X, W, and L as W(F) = XWL. Where L is a column vector expressing the characteristic of vertices in the molecule; X is a row vector expressing the bonding characteristics between adjacent atoms; W is a reciprocal distance matrix. The topological index W(F), based on the distance-related matrix of a molecular graph, is used to code the structural environment of each atom type in a molecular graph. The good QSPR/QSAR models have been obtained for the properties such as standard formation enthalpy of inorganic compounds and methyl halides, retention indices of gas chromatography of multiple bond-containing hydrocarbons, aqueous solubility, and octanol/water partition of benzene halides. These models indicate that the idea of using multiple matrices to define the modified Wiener index is valid and successful.     

A novel PI index and its applications to QSPR/QSAR studies

A novel topological index, PI (Padmakar-Ivan index), is derived in this paper. The index is very simple to calculate and has disseminating power similar to that of the Wiener (W) and the Szeged (Sz) indices. The comprehensive studies show that the proposed PI index correlates highly with W and Sz as well as with physicochemical properties and biological activities of a large number of diversified and complex compounds. The proposed PI index promises to be a useful parameter in the QSPR/QSAR studies. The stability of each model is demonstrated by applying cross-validation test. Furthermore, more favorable comparison with other representative indices such as the Randic index is also made in order to establish the predictive ability of the PI index. The results have shown that in several cases the PI index gave better results.     

直链烷烃取代衍生物Wiener指数的简便计算方法

根据直链烷烃衍生物分子拓扑结构的特点,将直链烷烃衍生物拆分为由直链单元和取代基团几个部分构成,再根据每部分的拓扑结构特点给出相应的计算公式.从而提出了一个计算直链烷烃衍生物Wiener指数的简便方法,达到简化计算的目的.该方法简化了传统Wiener指数的计算方法,使Wiener指数的计算具有效率高、不易出错等优点,便于Wiener指数计算程序化,从而提高了Wiener指数的实用性.     

Modification of Wiener Index and its application

A novel topological index based on the Wiener Index is proposed as W* = 1/2 sigma (n)(i,j=1) S(*)(ij), the element S(*)(ij) of the distance matrix is defined either as S(*)(ij) = alpha x square root of I(i)I(j)/R(ij) (atoms i and j are adjacent) or as S(*)(ij) = = alpha x (j-i+1)square root of I(i) x x x x x I(j)/R(ij) (atoms i and j are not adjacent), where I(i) and I(j) represent the electronegativity of vertices i or j, respectively, R(ij)() is the sum of the bond length between the vertices i and j in a molecular graph, and alpha = (Z(i)/Z(j))(0.5), where Z(i) and Z(j) are the atomic numbers of the positive valence atom i and the negative valence atom j, respectively. The properties and the interaction of the vertices in a molecule are taken into account in this definition. That is why the application of the index W to heteroatom-containing and multiple bond organic systems and inorganic systems is possible. Correlation coefficients above 0.97 are achieved in the prediction of the retention index of gas chromatography of the hydrocarbons, the standard formation enthalpy of methyl halides, halogen-silicon, and inorganic compounds containing transition metals.     

Topological characterization of cyclic structures

The sum of the topological distances in the molecular graph (the Wiener number) is used for a topological characterization of the condensed polycyclic molecular systems. This topological index discriminates well the isomeric cyclic molecules. In addition, it also properly reflects their structural features. On this basis the principal points of molecular cyclicity are formulated in 15 rules.     

Trees with the same Topological Index JJ

Abstract

In the present paper we investigate the trees with the same JJ index (called JJ-equivalent trees). The topological index JJ is derived from the so called Wiener matrix introduced by Randic et al., in 1994. The Wiener matrix is constructed by generalizing the procedure of Wiener for evaluation of Wiener numbers in alkanes. From such matrices several novel structural invariants of potential interest in structure-property studies were obtained. These include higher Wiener numbers, Wiener sequences, and hyper-Wiener number, etc. The topological index JJ is constructed by considering the row sums of the Wiener matrix. A construction method for a class of JJ-equivalent trees is given. By using this method we construct the smallest pairs of non-isomorphic JJ-equivalent trees which have 18 vertices. In addition we report on groups of 3,4, and 6 non-isomorphic JJ-equivalent trees. The smallest such trees are of size 28 for triples and quadruples, and 34 for the group of 6 elements.     

From Wiener index to molecules

In this paper we present an algorithm for the generation of molecular graphs with a given value of the Wiener index. The high number of graphs for a given value of the Wiener index is reduced thanks to the application of a set of heuristics taking into account the structural characteristics of the molecules. The selection of parameters as the interval of values for the Wiener index, the diversity and occurrence of atoms and bonds, the size and number of cycles, and the presence of structural patterns guide the processing of the heuristics generating molecular graphs with a considerable saving in computational cost. The modularity in the design of the algorithm allows it to be used as a pattern for the development of other algorithms based on different topological invariants, which allow for its use in areas of interest, say as involving combinatorial databases and screening in chemical databases.     

Molecular topological index: An application in the QSAR study of toxicity of alcohols

An application of the molecular topological index, adapted for heterosystems, in the QSAR study of toxicity of alkyl alcohols on fathead minnows (Pimephates promelas) is examined. The obtained QSAR model is comparable to models based on the Wiener number and the connectivity index.     

On structural interpretation of several distance related topological indices.

We consider the role that individual bonds play in bond-additivities in order to better understand the structural basis of various topological indices. In particular we consider indices closely related to the Wiener index (W) and the distance matrix and search for optimal weights of terminal and interior CC bonds in alkanes for a selection of physicochemical properties. It is interesting to note that different properties are associated with different relative roles of the exterior and the interior CC bonds.     

On the relation between W'/W index, hyper-Wiener index, and Wiener numbers

It is shown analytically that the W'/W index, the hyper-Wiener index, and the Wiener number are closely related graph-theoretical invariants for acyclic structures. A general analytical expression for the hyper-Wiener index of a tree is derived too.     

Topological Estimation of Aromatic Stabilities of Polyacenes and Helicenes: Modeling of Resonance Energy and Benzene Character

The distance-based topological indices viz Wiener (W)-, Szeged (Sz)-, Padmakar-Ivan (PI)- and Sadhana (Sd)-indices have been used for estimating aromatic stabilities as well as % benzene character of polyacenes and helicenes. Excellent models are obtained in combined set and there was no need to split the data set into polyacenes and helicenes. The newly introduced Sd index was found the best index among all the four indices used. In bi-parametric modeling with the combination of the Sd index with the PI index yielded excellent results. The results are discussed critically on the basis of variety of statistical parameters.     

Statistical properties of carbon nanostructures

We look at modeling carbon nanostructures from a theoretical graph network view, where a graph has atoms at a vertex and links represent bonds. In this way, we can calculate standard statistical mechanics functions (entropy, enthalpy, and free energy) and matrix indices (Wiener index) of finite structures, such as fullerenes and carbon nanotubes. The Euclidean Wiener index (topographical index) is compared with its topological (standard) counterpart. For many of these parameters, the data have power law behavior, especially when plotted versus the number of bonds or the number of atoms. The number of bonds in a carbon nanotube is linear with the length of the nanotube, thus enabling us to calculate the heat of formation of capped (5,5) and (10,10) nanotubes. These properties are determined from atomic coordinates using MATLAB routines.     

广义α~N指数及其应用 IV: 广义α~N指数在碳氢化合物中的应用

本文计算了七对顺、反异构体的广义α~N指数, 结果表明, 广义α~N指数不仅能很好地区分碳氢化合物的顺、反异构体, 而且与其物理性质(沸点、密度、折光率、分子折射度等)有良好的单变量线性相关关系, 解决了拓扑指数不能区分顺、反异构体的困难。本文还计算了庚烷、辛烷系列的广义α~N指数, 并比较了多种拓扑指数与辛烷值的相关分析结果。     

GTI-space: the space of generalized topological indices

A new extension of the generalized topological indices (GTI) approach is carried out to represent “simple” and “composite” topological indices (TIs) in an unified way. This approach defines a GTI-space from which both simple and composite TIs represent particular subspaces. Accordingly, simple TIs such as Wiener, Balaban, Zagreb, Harary and Randić connectivity indices are expressed by means of the same GTI representation introduced for composite TIs such as hyper-Wiener, molecular topological index (MTI), Gutman index and reverse MTI. Using GTI-space approach we easily identify mathematical relations between some composite and simple indices, such as the relationship between hyper-Wiener and Wiener index and the relation between MTI and first Zagreb index. The relation of the GTI-space with the sub-structural cluster expansion of property/activity is also analysed and some routes for the applications of this approach to QSPR/QSAR are also given.