On molecular topological properties of hex‐derived networks

Topological indices are numerical parameters of a molecular graph, which characterize its topology and are usually graph invariant. In quantitative structure–activity relationship/quantitative structure–property relationship study, physico‐chemical properties and topological indices such as Randić, atom–bond connectivity (ABC), and geometric–arithmetic (GA) index are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this area of research. In this paper, we study hex‐derived networks HDN1(n) and HDN2(n), which are generated by hexagonal network of dimension n and derive analytical closed results of general Randić index R_α(G) for different values of α, for these networks of dimension n. We also compute the general first Zagreb, ABC, GA, ABC₄, and GA₅ indices for these hex‐derived networks for the first time and give closed formulae of these degree‐based indices for hex‐derived networks. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

On the geometric–arithmetic index by decompositions-CMMSE

The concept of geometric–arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. There are many papers studying different kinds of indices (as Wiener, hyper–Wiener, detour, hyper–detour, Szeged, edge–Szeged, PI, vertex–PI and eccentric connectivity indices) under particular cases of decompositions. The main aim of this paper is to show that the computation of the geometric-arithmetic index of a graph G is essentially reduced to the computation of the geometric-arithmetic indices of the so-called primary subgraphs obtained by a general decomposition of G. Furthermore, using these results, we obtain formulas for the geometric-arithmetic indices of bridge graphs and other classes of graphs, like bouquet of graphs and circle graphs. These results are applied to the computation of the geometric-arithmetic index of Spiro chain of hexagons, polyphenylenes and polyethene.     

CoM-polynomial and topological coindices of hyaluronic acid conjugates

The molecular structure of corresponding drugs can be examined using a graph theory tool called topological index to learn about their physicochemical and biological properties. Topological index considers the pair of vertices that are connected while topological coindex takes into account the pairs of vertices that are not connected. Many of these topological indices can be readily calculated using various polynomial available in literature. We employ the concept of CoM-polynomial in this study and analyze the structure of hyaluronic acid conjugated with curcumin, paclitaxel, and methotrexate to acquire it. Many conventional topological coincides such as 1st Zagreb coindex, 2nd Zagreb coindex, 2nd modified Zagreb coindex, redefined 3rd Zagreb coindex, forgotten topological coindex, Randi? coindex etc. are also generated for these drugs.     

Computing weighted Szeged and PI indices from quotient graphs

The weighted (edge-)Szeged index and the weighted (vertex-)PI index are modifications of the (edge-)Szeged index and the (vertex-)PI index, respectively, because they take into account also the vertex degrees. As the main result of this article, we prove that if G is a connected graph, then all these indices can be computed in terms of the corresponding indices of weighted quotient graphs with respect to a partition of the edge set that is coarser than the Θ^*-partition. If G is a benzenoid system or a phenylene, then it is possible to choose a partition of the edge set in such a way that the quotient graphs are trees. As a consequence, it is shown that for a benzenoid system, the mentioned indices can be computed in sublinear time with respect to the number of vertices. Moreover, closed formulas for linear phenylenes are also deduced.     

Mostar indices of carbon nanostructures and circumscribed donut benzenoid systems

On the great success of bond-additive topological indices such as Szeged, Padmakar-Ivan, Zagreb, and irregularity measures, yet another index, the Mostar index, has been introduced recently as a quantitative refinement of the distance nonbalancedness and also a peripherality measure in molecular graphs and networks. In this direction, we introduce other variants of bond-additive indices, such as edge-Mostar and total-Mostar indices. The present article explores a computational technique for Mostar, edge-Mostar, and total-Mostar indices with respect to the strength-weighted parameters. As an application, these techniques are applied to compute the three indices for the family of coronoid and carbon nanocone structures.     

Calculation of retention indices by molecular topology. Chlorinated benzenes

A comparative study was undertaken to test the ability of several different topological indices to predict the retention indices of chlorinated benzenes on polar and non-polar stationary phases using both correlation coefficients and correctly predicted elution sequences as criteria of fit. The test was performed on three topological indices: connectivity indices, Wiener numbers, and Balaban indices. The regression analyses showed that the molecular connectivity model predicted the retention indices of chlorinated benzenes more successfully than either Wiener numbers or Balaban indices. The results also demonstrated that the major structural property controlling chromatographic behavior was the size of the chlorinated benzene. In addition, the use of the new non-empirical heteroatom parameterization scheme in the calculation of Wiener numbers and Balaban indices was successfully tested for the first time.     

A comparative QSAR study using Wiener, Szeged, and molecular connectivity indices

In this study we have investigated the relative correlation potential of Wiener (W), Szeged (Sz), and molecular connectivity indices (0chiR, 1chiR and 2chiR) in developing quantitative structure-activity relationships, QSAR; log P values of benzoic acid and its nuclear-substituted derivatives were used for this purpose. The statistical analyses for univariate and multivariate correlations had indicated that both W and Sz are closely related to the connectivity indices (mchiR) and that the W, the Sz, and the 1chiR indices have similar modeling potentials. 1chiR gives slightly better results than both W and Sz. Other connectivity indices 0chiR and 2chiR correlate poorly with log P.     

The Wiener distance matrix for acyclic compounds and polymers

The relationship between the Wiener indices and the topological structures of alkanes is analyzed. The expressions for the Wiener distances between elements of these structures are derived, and the distance matrix is constructed for them; this matrix is naturally called the Wiener distance matrix. The expressions for the Wiener indices of polymers with units of arbitrary structure are obtained.     

环烷烃及烷烃的折光指数与结构的关系研究

A new structure information autocorrelation topological index tX is designed and developed based on the vertex degree of molecular topology and autocorrelation function of mathematics. Quantitative structure property the relationships for estimating the refractive index of cycloalkane and alkane are set up based on multiple linear regression. The vertex degree is defined as βi .The structure information autocorrelation topological index tX is set up with the βi. The refractive index （nD）, for the 64 cycloalkanes, are correlated with this topological indices. The index, for the 27 alkanes, are also correlated with this topological indices. The calculated results showed that the calculated refractive index of cycloalkanes and alkanes are in good agreement with the experimental data, with the mean velative deviation 0.25%. With the established model, the refractive index of the other 5 alkanes are predicted.     

The overall Wiener index--a new tool for characterization of molecular topology.

Recently, the concept of overall connectivity of a graph G, TC(G), was introduced as the sum of vertex degrees of all subgraphs of G. The approach of more detailed characterization of molecular topology by accounting for all substructures is extended here to the concept of overall distance OW(G) of a graph G, defined as the sum of distances in all subgraphs of G, as well as the sum of eth-order terms, (e)OW(G), with e being the number of edges in the subgraph. Analytical expressions are presented for OW(G) of several basic classes of graphs. The overall distance is analyzed as a measure of topological complexity in acyclic and cyclic structures. The potential usefulness of the components of this generalized Wiener index in QSPR/QSAR is evaluated by its correlation with a number of properties of C3-C8 alkanes and by a favorable comparison with models based on molecular connectivity indices.