Optimal characterization of structure for prediction of properties

Different topological and physicochemical parameters have been used to predict hydrophobicity (logP, octanol-water) of chemicals. We calculated a hydrogen bonding parameter (HB₁) and a large number of molecular connectivity and complexity indices for a diverse set of 382 molecules. It is known from earlier studies that topological indices (TIs) predict properties of congeneric sets reasonably well. Since HB₁ is an approximate quantifier of hydrogen bonding and has integral values, we used HB₁ to classify the diverse set into strongly and weakly hydrogen bonding subsets. In an attempt to examine the utility of Us in predicting properties of relatively similar groups of molecules, we carried out a correlation of logP with TIs for a subset (n = 139) of the original diverse set (n = 382) with a weak hydrogen bonding ability (HB₁ = 0). Results show that TIs give a better predictive model for the more homogeneous subset as compared to the diverse set of molecules.     

Novel Indices for the Topological Complexity of Molecules

Abstract

The development of molecular complexity measures is reviewed. Two novel sets of indices termed topological complexities are introduced proceeding from the idea that topological complexity increases with the overall connectivity of the molecular graph. The latter is assessed as the connectivity of all connected subgraphs in the molecular graph, including the graph itself. First-order, second-order, third-order, etc., topological complexities ⁱ TC are defined as the sum of the vertex degrees in the connected subgraphs with one, two, three, etc., edges, respectively. Zero-order complexity is also specified for the simplest subgraphs–the graph vertices. The overall topological complexity TC is then defined as the sum of the complexities of all orders. These new indices mirror the increase in complexity with the increase in the number of atoms and, at a constant number of atoms, with the increase in molecular branching and cyclicity. Topological complexities compare favorably to molecular connectivities of Kier and Hall, as demonstrated in detail for the classical QSPR test-the boiling points of alkanes. Related to the wide application of molecular connectivities to QSAR studies, a similar importance of the new indices is anticipated.     

Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges

Research on the topological indices based on end-vertex degrees of edges has been intensively rising recently. Randić index, one of the best-known topological indices in chemical graph theory, is belonging to this class of indices. In this paper, we introduce a novel topological index based on the end-vertex degrees of edges and its basic features are presented here. We named it as geometrical-arithmetic index (GA).     

Chemical graphs with degenerate topological indices based on information on distances

Recently, four new types of vertex invariants, namelyu, v, x, andy, were defined on the basis of information on graph distances. They were combined to give four highly selective topological indices:U, V, X, andY. The degeneracy, i.e. equal values for nonisomorphic graphs, of the four topological indices is investigated. A structural condition and a graphical method which gives pairs of molecular graphs with identicalU, V, X, andY topological indices are introduced. The smallest pair of 4-trees representing alkanes having degeneratedU, V, X, andY values consists of trees with eighteen vertices.     

Molecular modeling of wine polyphenols

Due to wide range of health effects of wine polyphenols, it is important to investigate the relationship between their structure and physical properties (quantitative structure–property relationship, QSPR). We have investigated linear, nonlinear (polynomial), and multiple linear relationships between given topological indices and molecular properties of main pharmacological active components of wine, such as molecular weight (MW), van der Waals volume (Vw), molar refractivity (MR), polar molecular surface area (PSA) and lipophilicity (log P). Partition coefficient (log P) was calculated using three different computer program (CLOGP, ALOGPS and MLOGP). The best models were achieved using the MLOGP program. Topological indices used for correlation analysis include: the Wiener index, W(G); connectivity indices, χ(G); the Balaban index, J(G); information-theoretic index, I(G); and the Schultz index, MTI(G). QSPR was performed on the set of 19 polyphenols and, particularly, on the group of phenolic acids, and on the group of flavonoids with resveratrol. The connectivity index has been successfully used for describing almost all parameters. Significant correlations were achieved between the Wiener index and van der Waals volume, as well as molecular weight.     

Multivariate Association of Graph-Theoretic Variables and Physicochemical Properties

Abstract

We used canonical correlation analysis to examine the multivariate association between two distinct data sets commonly measured or calculated for approximately 600 chemicals: (1) measured or calculated values of select physieochemical properties (i.e., K _ow, boiling point, heat of vaporization, molecular weight, water solubility, molecular volume, hydrogen bonding potential, and vapor pressure) and (2) calculated algorithmically-derived variables (i.e., topological and neighborhood indices derived from graph theory). Canonical correlation analysis identified eight highly significant associations between linear combinations of graph-theoretic variables and linear combinations of physicochemical properties. The set of graph theoretic variables was significantly related to all physieochemical properties, explaining 55% to 99% of the variation in these properties.     

Discrimination of isomeric structures using information theoretic topological indices

Three newly defined information theoretic topological indices, namely “degree complexity (I^d),” “graph vertex complexity (H^V),” and “graph distance complexity (H^D)” along with three other information indices have been used to study their discriminating power of 45 trees and 19 monocyclic graphs. It is found that the newly defined indices have satisfactory discriminating power while H^D has been found to be the only index to discriminate all the graphs studied.     

New vertex invariants and topological indices of chemical graphs based on information on distances

By applying information theory to the set of topological distances from one vertex to all other graph vertices, one obtains four new types of vertex invariants (u _i,v _i,x _i,Y _i) which are real numbers (as opposed to integers). They may be combined in many ways to afford new topological indices. One such type leads to indicesU, V, X andY which show no degeneracy for alkanes with up to 15 vertices.     

Integration of Graph Theory and Quantum Chemistry for Structure-Activity Relationships

Abstract

The objective of this article is to outline both graph-theoretically based and quantum chemically based structural indices of potential use in quantitative structure activity correlations. We consider graph-theoretical indices such as the connectivity index, topological index, Wiener index and molecular ID indices. Several structural and geometry-dependent indices can be derived from semiempirical and ab initio quantum calculations based on the charge densities, overlap matrices, frontier orbitals, molecular hardness, free valence, density matrices, quantum spectral difference indices, quantum spectral indices and bond matrices. Finally, the use of electrostatic potentials and charge densities for the prediction of reactive sites will be discussed.     

The connective eccentricity index of graphs and its applications to octane isomers and benzenoid hydrocarbons

The connective eccentricity index (CEI) of a graph G is defined as , where ε_G(.) denotes the eccentricity of the corresponding vertex. The CEI obligates an influential ability, which is due to its estimating pharmaceutical properties. In this paper, we first characterize the extremal graphs with respect to the CEI among k-connected graphs (k-connected bipartite graphs) with a given diameter. Then, the sharp upper bound on the CEI of graphs with given connectivity and minimum degree (independence number) is determined. Finally, we calculate the CEI of two sets of molecular graphs: octane isomers and benzenoid hydrocarbons. We compare their CEI with some other distance-based topological indices through their correlations with the chemical properties. The linear model for the CEI is better than or as good as the models corresponding to the other distance-based indices.     

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.     

Molecular Graph Matrices and Derived Structural Descriptors

Abstract

A number of recently graph matrices encoding in various ways the topological information is presented. Four graph operators are used to compute 19 topological indices for a set of 306 alkanes. The intercorrelation coefficients of the 19 topological indices are computed and used to identify highly intercorrelated indices.     

Chemical graphs

In order to find the centre of an acyclic connected graph (of a tree), vertices of degree one (endpoints) are removed stepwise. The numbers _i of vertices thus removed at each step form a digit sequenceS (pruning sequence) which reflects the branching of the tree. The sum of squares of digits in the sequenceS affords a new topologicalcentric index B = _{i i} ² for the branching of trees. Comparisons with other topological indices are presented evidencing thatB induces an ordering of isomeric trees distinct from those induced by all other indices devised so far, becauseB emphasizes equally branches of similar length.It is shown that Rouvray's indexIis equivalent to Wiener's indexw, and that the Gordon-Scantlebury indexN ₂ and Gutmanet al.'s indexM ₁ belong to the same family, calledquadratic indices, and induce the same ordering.Since all topological indices vary both with the branching and the number of vertices in the tree, four new indices are devised fromB andM ₁ to account only (or mainly) for the branching, by normalization (imposing a lower bound equal to zero for chain-graphs, i.e.n-alkanes) or binormalization (same lower bound, and upper bound equal to one for star-graphs). Normalized and binormalized centric (C, C) and quadratic indices (Q, Q) are presented for the lower alkanes. From the five new topological indices, the centric indices (B, C, C) are limited to trees, but the quadratic indices (Q, Q) apply to any graph. Binormalized indices (C,Q) express the topological shape of the graph.     

Relation between geometric–arithmetic and arithmetic–geometric indices

The paper is concerned with the two topological indices, GA and AG, constructed from the geometric and arithmetic means of the end-vertices of edges. Inequalities involving GA and AG are established. The two indices are linearly correlated, whereas for molecular graphs of benzenoid and similar polycyclic conjugated systems, the linear relation between GA and AG is exact. This indicates that introducing (in 2015) the index AG, after GA has already been conceived (in 2009), was fully unjustified.

     

Generalized Hosoya polynomials of hexagonal chains

Similar to the well-known Wiener index, Eu et al. [Int. J. Quantum Chem. 106 (2006) 423–435] introduced three families of topological indices including Schultz index and modified Schultz index, called generalized Wiener indices, and gave the similar formulae of generalized Wiener indices of hexagonal chains. They also mentioned three families of graph polynomials in x, called generalized Hosoya polynomials in contrast to the (standard) Hosoya polynomial, such that their first derivatives at x = 1 are equal to generalized Wiener indices. In this note we gave explicit analytical expressions for generalized Hosoya polynomials of hexagonal chains.