Topological analysis of some special graphs. III. Regular polyhedra

A regular polyhedron is isomorphic to a cluster on which every face has same number of bonds and every atom has an equal number of coordinating atoms. A general strategy for generating the eigenvectors and the eigenvalues of regular polyhedra is given. Net sign analyses are also performed on the eigenvectors of regular polyhedra. The results provide us a quick way to grasp the topological feature of the electronic structure of clusters having interesting topology.     

Topological analysis of some special graphs: III. Regular polyhedra

A regular polyhedron is isomorphic to a cluster on which every face has same number of bonds and every atom has an equal number of coordinating atoms. A general strategy for generating the eigenvectors and the eigenvalues of regular polyhedra is given. Not sign analyses are also performed on the eigenvectors of regular polyhedra. The results provide us a quick way to grasp the topological feature of the electronic structure of clusters having interesting topology.     

On eigenvalues and eigenvectors of graphs

It is known that there exists an equivalence relation between the adjacency matrix of graph theory and the Hückel matrix of Hückel molecular orbital theory. This paper presents some useful methods which allow us to systematically find eigenvalues and eigenvectors of various classes of graphs without calculating characteristic polynomials. Results obtained from this study give insight into the topological studies of molecular orbitals.Dedicated to Professor Frank Harary on the occasion of his 70th birthday.     

Topological analysis of some special classes of graphs. II. steps,ladders, cylinders

A general strategy is proposed for generating the eigenvectors and the eigenvalues of some special classes of graphs from the well-known chemical graphs such as lines and cycles which are isomorphic to the hydrogen-suppressed linear and cyclic polyenes. This method is applied to step graphs, ladders, cylinders, etc. Net sign analyses are then performed for all these special classes of graphs.     

Close-to-zero eigenvalues of the rooted product of graphs

Journal of Mathematical Chemistry - The construction of vertex-decorated graphs can be used to produce derived graphs with specific eigenvalues from undecorated graphs, which themselves do not have...     

Comparability graphs and molecular properties: IV. Generalizations and applications

The development of a recently proposed method for calculating molecular properties is outlined. The approach is based on the idea of constructing optimized compound samples for structure—property or structure-activity correlations by means of the so—called comparability graphs (CG) of isomeric compounds. A dynamic comparability principle is devised, proceeding from a series of standard molecular rearrangements described in graph—theoretical terms as rules on molecular branching and cyclicity. An extension of the approach is presented for both the construction of CG's and their combination for variable numbers of atoms. The method is applied to various physico-chemical properties, which are thus divided into three groups according to the degree to which they are conditioned by molecular topology. The Wiener topological index is shown to produce a highly linear correlation with the alkane critical densities and volumes, as well as with their heats and entropies of vaporization.Dedicated to the memory of Professor Oskar E. Polansky, a pioneer of chemical graph theory     

Symmetry properties of chemical graphs. IV. Rearrangement of tetragonal-pyramidal complexes

Using a canonical numbering of vertices for the graph corresponding to a particular rearrangement of tetragonal–pyramidal complexes, all 120 permutations defining the symmetry for the rearrangement are derived. An examination of the permutations points to the symmetric group S₅, which has previously been found for isomerizations of trigonal–bipyramidal complexes and in the rearrangement of homotetrahedryl cations.     

Pyridazines. IV. The synthesis and some reactions of acetylpyridazines

Two 3-acetylpyridazines have been prepared. N-Oxidation of 3-acetylpyridazine ( 6 ) gave only 3-acetylpyridazine 1-oxide ( 7 ). During the N-oxidation of 3-acetyl-6-methoxypyridazine ( 10 ), three primary products, namely, 3-acetyl-6-methoxypyridazine 1-oxide ( 12 ), 3-acetyl-6-methoxy-pyridazine 2-oxide ( 13 ), 3-acetylpyridazin-6-one ( 14 ) and an artifact, 3-methoxypyridazine 1-oxide ( 15 ) were obtained. Furthermore, it has been shown that 3-methoxypyridazine 1-oxide ( 15 ) can be obtained in quantitative yield by treatment of 3-acetyl-6-methoxypyridazine 2-oxide ( 13 ) with dilute sodium hydroxide solution at room temperature. This represents a novel deacylation reaction. Nitration of 3-acetylpyridazine 1-oxide, ( 7 ) gave 3,4-bis(3′-pyridazinoyl)furoxan 1′,1′-dioxide ( 19 ) rather than a simple nitration product. 3-Acetyl-pyridazine ( 6 ) and 3-acetyl-6-methoxypyridazine ( 10 ) also gave furoxans ( 22 and 23 ) upon nitration.     

Wiener polynomials of some chemically interesting graphs

The chemist Harold Wiener found ??(G), the sum of distances between all pairs of vertices in a connected graph G, to be useful as a predictor of certain physical and chemical properties. The q‐analogue of ??, called the Wiener polynomial ??(G; q), is also useful, but it has few existing useful formulas. We will evaluate ??(G; q) for certain graphs G of chemical interest. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Tandem mass spectrometry and accurate mass analysis of some N-arylphthalimides. IV

The mass spectral studies of six N-arylphthalimides are reported. The low resolution spectra in conjunction with tandem mass spectrometry (MS/MS) and accurate mass measurements have provided valuable information, and established the fragmentation modes of the title compounds more precisely.     

Organotellurium derivatives : IV. Synthesis of some diaryltellurium dihalides and pseudohalides

Iodine monohalides (ICl, IBr), cyanogen halides (CNBr, CNI) and thiocyanogen (SCN)₂ add oxidatively to diaryltellurium(II), R₂Te (R = Ph, p-Ch₃OC₆H₄) under mild conditions ( °C). The resulting organotellurium(IV) derivatives react metathetically with silver pseudohalides to yield several new diaryltellurium(IV) pseudohalide derivatives.     

A graph-theoretical method for stepwise factorisation of symmetric graphs for simultaneous determination of eigenvectors and eigenvalues

A simple pictorial algorithm for factorisation of symmetric chemical graphs (weighted and unweighted) leading to simultaneous determination of their eigenvalues and eigenvectors has been devised. The method does not require group-theoretical techniques (viz. identification of the point group of the species under study, formation of symmetryadopted linear combinations using character tables etc.). It requires consideration of only one symmetry element, e.g., a reflection plane and is based on elementary row and column operations which keep the secular determinant of the adjacency matrix unchanged (except possibly for a multiplicative constant).     

Synthesis and molecular structures of some new titanium(IV) aryloxides.

The coordination chemistry of model phenolic ligands (pyrocatechol, salicylic acid, and 2,2'-biphenol) that are able to form respectively five-, six-, or seven-membered rings with titanium(IV) alkoxides is investigated. With pyrocatechol, a polynuclear complex containing 10 Ti atoms was characterized with a not very common doubly bridging mu3-(O,O,O',O') coordination mode. With salicylic acid, a monomeric tris(chelate) complex was obtained. With 2,2'-biphenol, a polynuclear complex containing six Ti atoms was obtained showing both mu2-(O,O') and mu2-(O,O,O') coordination modes for the ligands. Intermolecular interactions in the solid state for these three new compounds are also quantitatively discussed using the partial charge model.     

Total domination and open packing in some chemical graphs

In this paper we obtain some bounds and exact values of the total domination numbers and open packing numbers for pyrene torus, hexabenzocoronene, H-phenylenic nanotube and H-napthelenic nanotube.     

Dyad algebra and multiplication of graphs. II. Undirected graphs

In paper I abstract vectors |e>, their adjoints <e|, dyads such as |e><e|, and abstract linear operators were related to graphs which are in general directed. With an Hermitian operator one gets equivalence classes of undirected graphs with or without loops and multi-lines. The present paper II gives rules for the multiplication of such graphs based on their underlying dyad algebra. The results may be used in the evaluation of the outcome of successive applications of operators for observables as in the case of the powers of a one-electron Hamiltonian in the method of moments, or in using projection operators, electron density, and the like.