Zlatko Mihalić

On the Harary index for the characterization of chemical graphs

A novel topological index for the characterization of chemical graphs, derived from the reciprocal distance matrix and named the Harary index in honor of Professor Frank Harary, has been introduced. The Harary index is not a unique molecular descriptor; the smallest pair of the alkane trees with identical Harary indices has been detected in the octane family. The use of the Harary index in the quantitative structure-property relationships is exemplified in modeling physical properties of the C2-C9 alkanes. In this application, the performance of the Harary index is comparable to the performance of the Wiener number.

The distance matrix in chemistry

The graph-theoretical (topological) distance matrix and the geometric (topographic) distance matrix and their invariants (polynomials, spectra, determinants and Wiener numbers) are presented. Methods of computing these quantities are discussed. The uses of the distance matrix in both forms and the related invariants in chemistry are surveyed. Special attention is paid to the 2D and 3D Wiener numbers, defined respectively as one half of the sum of entries in the topological distance matrix and the topographic distance matrix. These numbers appear to be very valuable molecular descriptors in the structure property correlations.

The Laplacian matrix in chemistry

The Laplacian matrix, its spectrum, and its polynomial are discussed. An algorithm for computing the number of spanning trees of a polycyclic graph, based on the corresponding Laplacian spectrum, is outlined. Also, a technique using the Le Verrier-Faddeev-Frame method for computing the Laplacian polynomial of a graph is detailed. In addition, it is shown that the Wiener index of an alkane tree can be given in terms of its Laplacian spectrum. Two Mohar indices, one based on the Laplacian spectrum of a molecular graph G and the other based on the Laplacian x2 eigenvalue of G, have been tested in the structure-property relationships for octanes.

The Detour Matrix in Chemistry

The detour matrix of a (chemical) graph is defined. The detour matrix is also defined for weighted graphs. A novel method of computing the detour matrix is introduced. Some properties of the detour matrix and the distance matrix are compared. The invariants of the detour matrix (detour polynomial, detour spectrum, and detour index) are discussed, and several methods for computing these quantities are presented. The use of the detour index is analyzed and compared to the application of the Wiener number in the structure−boiling point modeling.

On the geometric-distance matrix and the corresponding structural invariants of molecular systems

Abstract A recently introduced concept of the geometric-distance polynomial by Balasubramanian is used for differentiating conformational isomers. Related geometry-dependent invariants (the spectrum of the geometric-distance matrix and the three-dimensional Wiener number) are also used for the same purpose. The geometric-distance matrix is computed by means of a molecular-mechanics method, and its characteristic polynomial by a modified Le Verrier—Fadeev—Frame—Balasubramanian method.

Graphical bond orders: Novel structural descriptors

We outline an algorithm for construction of novel molecular descriptors from known structural invariants or molecular properties viewed as descriptors. The novel descriptors are bond-additive quantities derived by assigning to each bond a contribution x’obtained by evaluating invariant X for graph G’-e, which is attained from graph G (representing a given molecule) by deleting edge e. The molecular descriptor X’/X is obtained as a normalized sum of bond contributions, where X’is equal to the sum of bond orders x’. The approach is illustrated by presenting X’/X descriptors for smaller alkanes for several well-known topological indices, including the connectivity index, Hosoya’s Z index, the Wiener index, and others. The algorithm is quite general and allows one to include molecular properties as source data for the construction of novel descriptors. This is particularly important in view of a limited number of properties used as descriptors in traditional quantitative structure-activity studies. The new algorithm literally doubles thenumber of descriptors available to traditional chemometricians in their quest for novel property-activity relationships.

APPLICATION OF TOPOGRAPHIC INDICES TO CHROMATOGRAPHIC DATA : CALCULATION OF THE RETENTION INDICES OF ALKANES

Abstract Gas chromatographic (GC) data on alkanes have been re-examined from a structural point of view with emphasis on the distances in optimum (minimum-energy) confromations. The information on the distances in an alkane G is embodied in the topographic (geometric) distance matrix D (G) and the related 3-D Wiener number 3 W (G) of G. The optimum 3-D structures of alkanes were obtained from the molecular mechanics computations. The GC retention indices ( I ) of 157 alkanes were calculated using a three-parameter equation of the form I = a [ 3 W (G)] b + c . The calculated I values are in excellent agreement with the experimental values. A comparison between the reported results and those obtained with the 2-D Wiener number and the connectivity index is also discussed.

The algebraic modelling of chemical structures: On the development of three-dimensional molecular descriptors

Abstract A novel approach to algebraic modelling of molecular structures is proposed. Structures of molecules are characterized by a single number derived from the topographic (geometric) distance matrix of the molecule. The topographic distance matrix is obtained in the following way. The idealized structure of a molecule is set up and then processed by the molecular mechanics method. This computation gives the optimum (minimum-energy) molecular geometry which is used for computing the topographic distance matrix. The half-sum of the elements of the topographic distance matrix is named the three-dimensional (3-D) Wiener number because of its formal similarity with the twodimensional (2-D) Wiener number which is equal to the half-sum of elements in the graph-theoretical distance matrix. The 3-D Wiener number is used to build, as an illustrative example, the 3-D structure-property model for predicting boiling points of alkanes. The comparison between the models based on the 2-D and 3-D Wiener numbers and the connectivity index is also discussed.

Neighboring sulfur participation in the solvolysis of 2-(ω-alkylthioalkyl)-3-methyl-2-cyclohexenyl p-nitrobenzoates

Abstract As shown by kinetic and product analysis, the solvolysis of 7 and 8 in 97% TFE includes several competitive reactions, one of them being neighboring sulfur participation and formation of intermediate cyclic sulfonium cation.

Vibrational spectroscopic and DFT calculation studies of cobalt(II) complexes with 3-hydroxypicolinic acid.

Two cobalt(II) complexes with 3-hydroxypicolinic acid (3-hydroxypyridine-2-carboxylic acid, 3-OHpicH), trans-[Co(3-OHpic)2(py)2] (2) and cis-[Co(3-OHpic)2(4-pic)2] (3) (py=pyridine; 4-pic=4-picoline or 4-methylpyridine), previously synthesized and characterized by X-ray diffraction, are here studied by Raman and mid-infrared spectroscopy with the help from the corresponding DFT vibrational calculations using B3LYP/6-311G(d,p) computational model. Intramolecular O-H⋯O hydrogen bond appears in both complexes 2 and 3, while weak C-H⋯O hydrogen bonds assemble molecules of 2 or 3 into 3D architecture. A complete presentation of all Raman, infrared and theoretical results is given for complex 3. The measured spectra are shown, relative intensities and bandwidths are discussed and the assignment of vibrational bands is given on the basis of the DFT calculations. The calculated spectra agree very well with the presented experimental findings, thanks to the suitable grouping of modes. The same vibrational calculations also reveal insignificant influence of H→CH3 substitution for the spectroscopic characterization of the complex. A careful study of differences between calculated and observed wavenumbers suggests that modified single-factor scaling is actually better than the classic multi-factor scaling approach.