GrInvIn in a nutshell

GrInvIn (Graph Invariant Investigator) is a software framework for teaching graph theory and for research in graph theory and graph theoretic chemistry. It enables users to construct graphs, compute invariants (e.g. topological indices in chemistry) and investigate relations between these concepts. The design of GrInvIn emphasizes easy usage and makes use of software engineering techniques that enable the user to easily extend the system (e.g. by adding new topological indices to investigate).     

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.     

Molecular Similarity and Risk Assessment: Analog Selection and Property Estimation Using Graph Invariants

Abstract

Four molecular similarity measures have been used to select the nearest neighbor of chemicals in two data sets of 139 hydrocarbons and 15 nitrosamines, respectively. The similarity methods are based on calculated graph invariants which include atom pairs, connectivity indices and information theoretic topological indices. The property of the selected nearest neighbor by each method was taken as the estimate of the property under investigation. The results show that for these data sets, all four methods give reasonable estimates of the properties studied.     

On graph isomorphism and graph automorphism

The problem of graph isomorphism, graph automorphism and a unique graph ID is considered. A new approach to the solution of these problems is suggested. The method is based on the spectral decompositionA = _{i i} K _iof the adjacency matrixA. This decomposition is independent of the particular labeling of graph vertices, and using this decomposition one can formulate an algorithm to derive a canonical labeling of the corresponding graphG. Since the spectral decomposition uniquely determines the adjacency matrixA and hence graphG, the obtained canonical labeling can be used in order to derive a unique graph ID. In addition, if the algorithm produces several canonical labelings, all these labelings and only these labelings are connected by the elements of the graph automorphism groupG. In this way, one obtains all elements of this group. Concerning graph isomorphism, one can use a unique graph ID obtained in the above way. However, the algorithm to decide whether graphG andG are isomorphic can be substantially improved if this algorithm is based on the direct comparison between spectral decompositions of the corresponding adjacency matricesA andA.Research supported by the Yugoslav Ministry for Development (Grant P-339).     

Die Konstruktion von elektronischen Superiorstrukturen

Summary Using the matrixC of a quantumchemical problem =C, hierarchic classification methods and the connex-tableau-algorithm [20, 21] based on graph theory, superior structure of an electronic system were constructed. These structures are fragments with maximal electronic similarity in relation to the information given byC. It is possible to study the relative stability and the inner structure of the superior fragments. The method is demonstrated for three hydrocarbons with symmetry C_1h using results of HMO calculations [24].

In memoriam Professor Dr. Oskar E. Polansky     

On physical analysis of entropy measures for sodium oxide via statistical models

This study delves into a comprehensive physical analysis of entropy measures applied to sodium oxide ${\text{Na}}_2\text{O}$. A key idea in information theory and thermodynamics, entropy is essential to comprehending the stability of a system. The study clarifies the intricate relationships and physical properties of ${\text{Na}}_2\text{O}$ by combining theoretical analysis with statistical methods, providing important knowledge for material design and industrial operations. In a chemical graph, atoms are shown by vertices while their bonding are illustrated by edges. Sodium oxide is a major contributor to manufacture glass, it also has potential applications in ${\text{CO}}_2$ sequestration, transparent materials, biomedical devices and nano grating glass. A topological index is a relationship between the molecular graph and its topology. We have computed various graph entropies based on different topological indices of chemical graph of sodium oxide. Further we have integrated these graph entropies with distinct thermodynamical measures of sodium oxide by developing mathematical models between both quantities. We have developed these mathematical frameworks in MATLAB. All the models are selected relying on the least mean squared error or sum of squared error.     

Description of oxygen permeability in various high polymers using a graph theoretical approach

Graph theory methods are shown to complement group additivity methods of predicting oxygen permeability in certain types of polymers. Graph theory is a topological approach that assigns a set of indices to a molecule to describe its structure. Since many physical properties of molecules depend upon their structure, graph theory indices can be used to describe important properties of molecules. In this work a set of graph theory indices are used to describe the property of a polymer based on a modified representation of the monomer unit. More specifically, Randic indices are used to describe the log of the oxygen permeability with 3.2% average relative error. Polymers comprising the basis set contain backbones of sp², sp³, or aromatic carbons, oxygen, or silicon and have substituents that contain chloride, fluoride, alkyl groups, hydrogen, oxygen, aromatic carbons, or chloro and/or fluoro substituted alkyl groups. The correlation coefficient (R²) (0 ≤ R² ≤ 1) of a nonlinear model is 0.91. The graph theory method for describing the oxygen permeability of these selected groups of polymers is in good agreement with that predicted by the permachor model. The permachor method makes oxygen permeability predictions based upon group additivity and distinguishes the degree of crystallinity of a polymer by empirically assigning different permachor (π) values to identical groups based upon the polymer crystallinity. The inability of graph theory to explain the remaining 9% of the scatter in the data is probably due to failure to incorporate into the graph theory model terms which quantify crystallinity.     

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.     

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.     

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.     

Resistance distance local rules

In [D.J. Klein, Croat. Chem. Acta. 75(2), 633 (2002)] Klein established a number of sum rules to compute the resistance distance of an arbitrary graph, especially he gave a specific set of local sum rules that determined all resistance distances of a graph (saying the set of local sum rules is complete). Inspired by this result, we give another complete set of local rules, which is simple and also efficient, especially for distance-regular graphs. Finally some applications to chemical graphs (for example the Platonic solids as well as their vertex truncations, which include the graph of Buckminsterfullerene and the graph of boron nitride hetero-fullerenoid B ₁₂ N ₁₂) are made to illustrate our approach.     

Graphical unitary group approach to arbitrary spin representations

A graph theoretic approach to the representation of N particle systems involving arbitrary single-particle spin is presented. The method is a generalization of the Distinct Row Table (DRT ) technique employed by Shavitt in developing the graphical unitary group approach (GUGA ) to electronic spin-orbitals. A detailed analysis of the DRT and GUGA representations is presented based on several theoretical considerations and computer-tested implementations. The use of the representation to establish rules concerning the evaluation of the matrix elements of the group generators is also discussed.     

Application of the Coupled Redox and Complexation Reactions to Flow Injection Spectrophotometric Determination of Promazine

Abstract

A flow injection (FI) spectrophotometric method is proposed for the determination of promazine hydrochloride. The method is based on the coupled redox - complexation reactions which proceed in the promazine-iron(III) and 1,10-phenantroline system. A linear calibration graph was obtained between 2–12 ppm of promazine hydrochloride with a sampling rate of 163 samples h^?1. The proposed method was applied to the determination of promazine in pharmaceuticals.     

Kinetic Spectrophotometric Determination of Manganese by Use of the Salicylaldehyde-Hydrogen Peroxide System

Abstract

A kinetic method is described for determining trace amounts of manganese(II), based on its catalytic effect on the oxidation of salicylaldehyde by hydrogen peroxide. The reaction is followed spectrophotometrically by measuring the rate of change of absorbance at 500 nm. The calibration graph is linear in the range 5–100 ng/ml with a relative error of ± 1.2%. The method has been applied to the determination of manganese in natural water.     

Treatment of Model Error in Calibration by Robust and Fuzzy Procedures

Abstract

In every mathematical (e.g., statistical) procedure and theorem used in calibration, several conditions need to be fulfilled. What can analysts and chemometricians do, however, if the conditions are only nearly fulfilled? One can expect that small changes in the conditions yield only small changes in the results. This article shows how to treat two types of model error caused by assuming an incorrect error distribution or relationship (i.e., linear). The procedures applied are based on robust statistics and fuzzy theory, respectively.     

WKB Level Spectra in Two Inhomogeneous Electron Liquids: Na Clusters and C Cages

Abstract

After a brief summary of known magic numbers and eigenvalue sums for (a) the hydrogenic potential—Ze ²/r and (b) the three-dimensional isotropic harmonic oscillator potential, approximate semi-classical scaling laws are presented relating to WKB eigenvalues for two more complex central potentials V(r) The first is the so-called Woods-Saxon potential, used in early work to calculate electronic magic numbers in clusters of Na atoms. The second potential V(r) chosen arises from a simple surface charge model of almost-spherical carbon cages such as C⁶⁰. For these last two potentials, semi-classical theory is shown to lead to qualitative insight, without very lengthy mathematical calculations.     

Extractive Spectrophotometric Determination of Ondansetron by Ion-Pair Formation with Bromocresol Green

Abstract

An empirical spectrophotometric procedure for the determination of the antiemetic ondansetron is carried out. The method is based on the formation of a 1:1 ion pair with bromocresol green in the pH range over 3.2 – 4.4, extraction into chloroform layer and spectrophotometric measurement at 420.8 nm. The calibration graph is linear over the range 0.1 – 20 μg ml^?1 ondansetron, with a relative standard deviation of 2.7%; the influence of foreign substances is also studied. The method is applied to ondansetron determination in human urine.     

Determination of Phenolic Compounds in Waste Waters by Sequential Injection Analysis and Spectrophometry

Abstract

A sequential injection analysis (SIA) system based on oxidative coupling of phenolic compounds with 4-aminoantipyrine (4-AAP) in alkaline solution and spectrophotometrical measuring of the absorbance at 510 nm has been developed for the determination of phenolic compounds in wastewaters. The proposed system is fully automatized and is able to monitor phenolic compounds in samples at a frequency of about 24 samples per hour with a relative standard deviation (RSD) better than 0.6%. The calibration graph is linear between 0.05 and 25 mg.dm^?3.