Graph Theoretical Indices Research Articles

Description of Vapour Pressures of Polycyclic Aromatic Compounds by Graph Theoretical Indices

AbstractThe vapour pressure (VP) is an important physical property of organic compounds for the description of fate and distribution of xenobiotics in the environment. This property was measured for 30 polycyclic aromatic hydrocarbons (PAH) and formulas for VP based on molecular weight and topological indices, respectively, were derived using the technique of Kubinyi and as a second step by regression analysis. A formula for the 30 PAH, considered here, is given.magnified imageTherein the VP is related to the information theoretical index IdWmean and with a chain‐connectivity index. More specific results are derived for three groups of PAHs. The robustness of the equations is tested by bootstrap analysis. In addition a formula is discussed, which is based on four different incremental factors, a method widely used in physical organic chemistry.

Extensions of the Wiener Number

Particularly for structure−property correlations there are many chemical graph-theoretic indices, one of which is Wiener's “path number”. Because Wiener's original work focused on acyclic structures, one can imagine different ways of extending it to cycle-containing structures, several of which are noted here. Many of these different formulas in fact yield like numerical values for general connected graphsthat is, different formulas sometimes correspond to the same graph invariant. Indeed it is found that there are two “dominant” classes of formulas each corresponding to one of two distinct invariants. Extensions to sequences of invariants (with the Wiener index the first member) are more often found to give distinct sequences. Further a powerful vector-space theoretic view for characterizing and for comparing different sequences is described. This is illustrated for a collection of eight sequences in application to a set of molecular graph structures (corresponding to the octanes). Another type of extension is to generate a sequence or partially ordered set of graph invariants for which the Wiener index is the natural first member of this set. Certain such sets of invariants (corresponding to contributions from different types of subgraphs) are noted to be “complete” (or form a basis) in the sense that any invariant can be faithfully linearly expanded in terms of the members of the set.

Geometry-dependent connectivity indices for the characterization of molecular structures

Geometry-dependent connectivity indices are proposed based on the Euclidian connectivity distances between the nuclei for characterizing the molecular structures. It is shown that these indices are directly related to the compactness of molecules in the three-dimensional space and thus these indices differentiate structures which otherwise may not be differentiated by graph-theoretical indices as exemplified with chrysene and benzo(c)phenanthrene.

Linear Correlation between Properties and Graph Theoretical Indices of Homologous Molecules

Linear Correlation between Properties and Graph Theoretical Indices of Homologous Molecules

Raoul Naroll's Contribution to Archaeology

Raoul Naroll's 1962 American Antiquity article on the floor area of dwellings and its relation to population has had a profound impact on archaeology. Not only did it provide an objective mechanism to estimate population size and hence to evaluate demographic models of cultural evolution, it also helped to spark a wide variety of cross-cultural investigations of settlement form and its relation to social, political, and economic organization. In this article, the author offers an example of one such study focusing on the permeability of communities. Permeability is the extent to which dwellings are embedded in a community, measured by a graph-theoretical index. A holocultural analysis is used to demonstrate a strong, positive relationship between frequency of war and community permeability. The author discusses the utility of identifying this relationship for archaeological research, particularly for evaluating conflict-oriented models of cultural evolution.

Integration of graph theory and quantum chemistry for structure-activity relationships.

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.

Correlation of graph-theoretical parameters with biological activity.

Ośmialowski and Kaliszan calculated graph-theoretical indices for substituted isonicotinic hydrazides and used simple and multiple regression to search (unsuccessfully) for correlations with biological activity. The present paper describes successful searches for correlation in the same data set using principal component analysis (PCA) with multivariate outlier testing and also using stepwise multiple regression. Following PCA, correlation with biological activity always appeared in the second principal component, not the first, that is, after projection of the data points into the (n - 1)-space orthogonal to the first principal component axis. In that space, the principal component score was a more accurate predictor of biological activity than were equations provided by multiple regression or stepwise multiple regression using the underlying variables. A multivariate outlier test identified one observation as discordant, and removing that observation improved prediction further.

Studies of Performance of Graph Theoretical Indices in QSAR Analysis

AbstractComparative studies of performance in exemplary QSAR of the most commonly used structural descriptors derived from molecular graphs were performed. Three diverse series of test compounds were selected for which various property measures existed. One group of compounds were polychlorinated dibenzofurans of determined gas chromatographic retention indices. The second group comprised isomers of general formula C4H8O of known boiling points, refractivity indices and densities. Third set of agents consisted of 2‐substituted isonicotinic acid hydrazides of determined antibacterial activity. The following graph theoretical indices were considered: the Kier and Hall [22] generalized molecular connectivity indices, the Balaban distance sum connectivity index calculated after Barysz et al. [23], the indices of molecular shape of Kier [24], the information indices of neighbourhood symmetry of Sarkar et al. [25] and of Basak et al. [26]. Auxiliary structural parameters, e.g., molecular surface area, were also employed. Significant correlation with molecular mass and intercorrelations among the graph theoretical indices were observed excepting the group of information content indices. For the compounds and properties studied the graph theoretical indices considered appeared unable to provide a size‐independent, meaningful input to property description.

Graph-theoretic cluster expansions. Thermochemical properties for alkanes

Numerical studies of Klein's graph-theoretic cluster expansion ansatz (CEA) are made for several thermochemical properties and graph-theoretic (“topologica”) indices of alkanes. The ansatz expresses any molecular property in terms of unique numerical contributions arising from structural descriptors, subgraphs of the molecular structural formula, called clusters. The collection of cluster contributions comprises a unique fingerprint or signature characterizing each property. Signatures are pattern recognition tools which can be used to (1) identify the most significant structural descriptors for each property, and (2) analyze for similarity and dissimilarity among properties and indices. Visual renderings of CEA signatures display the structural origins of properties in an easily appreciated form.

A graph-theoretic approach to predicting molecular properties

During the last two decades many graph invariants have been developed and used for predicting properties of molecules. However, no systematic study has been undertaken to test the effectiveness of these parameters in correlating properties of large and diverse groups of molecules. We have used a large set of graph invariants in the prediction of lipophilicity (log P, octanol-water) of a number of large and diverse databases of chemicals. The utility of graph theoretic indices in structure-activity relationship (SAR) and property-activity relationship (PAR) in general is discussed. Correlations of lipophilicity with calculated graph invariants is analyzed in terms of molecular factors that determine this property.

Graph-theoretical study of morphine-, morphinan-and benzomorphan-related opioids

A graph-theoretical method, based on the calculation of paths of different lengths, is applied to model the relationship between structure and biological activity in the group of morphine-, morphinan- and benzomorphan-related opioids. The structural properties of some groups of atoms as well as individual atoms within the molecules are studied and related to existing theories of drugreceptor interaction of the opiate drugs. The results indicate that it is possible to find a graph-theoretical description which could be useful in the structure-activity correlation investigations and the method seems to be very promising in the examination of interactions between drugs and their receptors. The graph-theoretical index, derived after examination of the compounds with known biological activity, is used for predicting the activity of other compounds which can be useful as novel drugs.

Correlation for kováts retention index of C 9C 26 mono-alkyl and polymethyl alkanes and alkenes

Correlations based on graph theoretic indices are proposed for prediction of the Kováts retention index of high-molecular-weight monoalkyl and polymethyl alkanes and alkenes. As a test for the proposed correlations, the Kováts index of monoalkyl hexadecanes in the C 18C 27 range were experimentally measured and compared with predicted values. Agreement between experimental and predicted values were quite satisfactory with an average error of 8 index units.

Molar volumes of alkanes and topological indices

This paper examines possible explanations for correlations that exist between graph-theoretical topological indices and observed physical properties of molecules. The general line of approach is to demonstrate that a particular graph-theoretical correlation has an analytical correspondence to an observed additive structural description. The basic approach is exemplified by showing that the correlation of partial molar volumes of normal alkancs with (chi) topological indices corresponds quantitatively to a previously obtained molecular structure relationship involving group additivity and average gauche conformation numbers. A recurrence relation is tested for the gauche numbers; and the results confirm a previously proposed linear dependence on molecular size.

A rational selection of graph-theoretical indices in the QSAR

In this paper we have addressed an important problem: how to develop a strategy for identification of the topological indices which are most important in establishing a model for structure–activity analysis. In particular we consider a rational selection of graph theoretical (topological) indices in QSAR. The approach is illustrated by the study of the toxicity of ethers on mice. The indices examined include the connectivity indices based on adjacency matrix, the conceptually related graph distance indices, and an information theoretic index based on partitioning of electrons in various domains. We have outlined the similarities and differences and discuss advantages and limitations among different approaches. Our emphasis is on the problem how to select in a rational way appropriate graph-theoretical descriptors and how to introduce higher “corrections” in order to improve derived correlation.

Determining structural similarity of chemicals using graph-theoretic indices

Ninety (90) graph-theoretic indices were calculated for a diverse set of 3692 chemicals to test the efficacy of using graph-theoretic indices in determining similarity of chemicals in a large, diverse data base of structures. Principal component analysis was used to reduce the 90-dimensional space to a 10-dimensional subspace which explains 93% of the variance. Distance between chemicals in this 10-dimensional space was used to measure similarity. To test this approach, ten chemicals were chosen at random from the set of 3692 chemicals and the five nearest neighbors for each of these ten target chemicals were determined. The results show that this measure of similarity reflects intuitive notions of chemical similarity.

Searching for the endogenous benzodiazepine using the graph theoretical approach.

The graph theoretical indices of several compounds with reported benzodiazepine receptor binding affinities were calculated. Our results demonstrate a structural similarity among diazepam, triazolam, and the beta-carboline nucleus and a structural dissimilarity to the purines and nicotinamide. This result correlates with their respective binding affinities. Using the graph theoretical indices as structural descriptors of the benzodiazepines and the significant ligands of the beta-carbolines, a search for peptide sequences as potential ligands was explored. Single amino acids through pentapeptides with all possible amino acid substitutions and chemical modifications were calculated. The peptides generated were subjected to graph theoretical analysis, and their indices were compared to those of the benzodiazepines. Comparisons resulted in seven dipeptides and six tripeptides that are topologically similar to the benzodiazepines and beta-carbolines. The dipeptides are histidine- or tryptophan-containing compounds with pyroglutamine, phenylalanine, and tyrosine residues in the second position. The tripeptides have two aromatic amino acid residues and a pyroglutamine or glycyl terminal residue. These structures are promising candidates because (1) they are structurally (topologically) similar to the benzodiazepines, represented by diazepam and triazolam, and to the beta-carbolines; and (2) they are sequences that may reasonably form a part of a larger peptide or that may be formed metabolically by proteolysis.

Quantitative structure-activity study on human pharmacokinetic parameters of benzodiazepines using the graph theoretical approach.

The graph theoretical indices for a series of 13 benzodiazepines were calculated using a graph-path topological method. The total molecule, the ring fragments, and combinations of ring fragments were subjected to a quantitative structure-activity analysis using eight pharmacokinetic parameters. The metabolic clearance and the blood-to-plasma concentration ratios were most highly correlated with the graph theoretical indices, with R values of 0.975 and 0.938, respectively. These correlations were found when the diazepine + benzo fragment and phenyl fragment were used to calculate the graph-path indices. Terminal disposition half-life was correlated with the benzo + diazepine fragment, with R = 0.969. Truncating the graph-path codes by eliminating cycles in the total molecule markedly improved the correlation coefficients. When compared to the graph-path indices for the total molecule, the correlation coefficients for the terminal disposition half-life and metabolic clearance data rose from 0.721 to 0.935 and from 0.770 to 0.968, respectively, using the graph-path indices of the truncated molecule. Intrinsic clearance of unbound drug also was poorly correlated with the total molecule (r less than 0.7) but rose significantly using the graph-path indices of the truncated molecule (r = 0.971 and 0.975 for the well-stirred and parallel-tube models, respectively).

Discrimination and ordering of chemical structures by the number of walks

The use of the number of walks for the discrimination of graphs representing chemical structures is discussed. A highly selective graph-theoretical index based on the number of walks is defined. The index values were calculated for 661 acyclic and 376 cyclic structures. The selectivity of the new index is compared to that of some of the most selective previously defined indexes. The ordering of structures induced by the value of the index is also considered.