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.     

Block renormalization group for Euclidean Fermions

Block renormalization group transformations (RGT) for lattice and continuum Euclidean Fermions in d dimensions are developed using Fermionic integrals with exponential and -function weight functions. For the free field the sequence of actionsD _k generated by the RGT from D, the Dirac operator, are shown to have exponential decay; uniform ink, after rescaling to the unit lattice. It is shown that the two-point functionD ^–1 admits a simple telescopic sum decomposition into fluctuation two-point functions which for the exponential weight RGT have exponential decay. Contrary to RG intuition the sequence of rescaled actions corresponding to the -function RGT do not have uniform exponential decay and we give examples of initial actions in one dimension where this phenomena occurs for the exponenential weight RGT also.     

Topological indices for molecular fragments and new graph invariants

Whereas the internal fragment topological index (IFTI) is calculated in the normal manner as for any molecule, the external fragment topological index (EFTI) is calculated so as to reflect the interaction between the excised fragment F and the remainder of the molecule (G-F). For selected topological indices (TIs), a survey of EFTI values, formulas and examples is presented. Some requirements as to the fragment indices are formulated and examined. In the discussion of the results, it is shown that for some TIs regularities exist in the dependence of EFTI values upon the branching of fragment F, or upon the marginal versus central position of the fragment F in the graph G. New vortex invariants can be computed as EFTI values for one-atom fragments over all graph vertices; by iteration, it is in principle possible to devise an infinite number of now vertex invariants.     

Unique description of chemical structures based on hierarchically ordered extended connectivities (HOC procedures). V. New topological indices,ordering of graphs,and recognition of graph similarity

The vertex numbering obtained by application of the HOC algorithm can be converted into two sequences of numbers: If each vertex starting with vertex 1 is only counted once, the sums of numberings of adjacent vertices form sequence S_i (i = 1?N), while the sums of S_i values form sequence M_i (i = 1?N). These two sequences can be used for (i) two new topological indices, ?? and ??, the latter being of extremely low degeneracy, and the former correlating with boiling points of alkanes; (ii) a criterion based on sequence S_i for ordering graphs which possess the same number N of vertices; and (iii) a quantitative measure, also based on sequence S_i, for appreciating the similarity or dissimilarity of pairs of graphs. Comparisons with other topological indices, ordering criteria, and similarity measures for graphs show that the newly devised procedures compare favorably with those known previously.     

The numbers of chiral and achiral alkanes and monosubstituted alkanes

Whereas the theory for the enumeration of the optical isomers of the lakyl radicals and the alkanes has long been understood, this is not the case for the corresponding archiral isomers. We present for the first time recurrence formulae for counting the number of archiral isomers of the alkyl radicals and the alkanes. For chiral and archiral alkanes and monosubstituted alkanes, numerical results up to C₁₄ are tabulated.After presenting the history of the problem and the necessary definitions, we proceed to derive functional equations on the various generating functions, which readily yield the more explicit recurrence formulae usefule for numerical calculations. In the process, we first re-derive Pólya's expression for planted steric trees using his classical enumeration theorem. This result is then extended to the enumeration of free steric trees using the now standard tree-counting method due to Otter and known as a dissimilarity characteristic equation.By definition, a steric tree is a quartic tree (all points having degree 1 or 4) in which the four neighbors of every carbon point are given a tetrahedral configuration. Building on the methods of the first two authors for counting chiral and archiral trees in the plane, we obtain the formula for counting achiral steric trees, thus setting a problem first enunciated by van't Hoff and Le Bel in 1874.     

NMR-Spektren vontrans—Chalcon-oxim,-semicarbazon und-thiosemicarbazon und ihren cyclischen Isomeren (3,5-Diphenyl-2-isoxazolin und 1-substituierte 3,5-Diphenyl-2-pyrazoline)

Zusammenfassung Frühere Strukturvorschläge für dastrans-Chalcon-oxim, semicarbazon und thiosemicarbazon¹ werden mit Hilfe von NMR-Spektren richtiggestellt: Die sauer katalysierte Reaktion führt zwar zum Oxim, Semicarbazon und Thiosemicarbazon destrans-Chalcons; basische Katalysatoren bewirken dagegen eine interne Cyclisierung zu 3,5-Diphenyl-2-isoxazolin bzw. zu 1-substituierten 3,5-Diphenyl-2-pyrazolinen. Die Zuordnungen in den NMR-Spektren wurden durch Vergleich mittrans-Dypnon und seinem Semicarbazon getroffen.

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NMR spectra of trans-chalcone oxime, semicarbazone and thiosemicarbazone and their cyclic isomers (3,5-diphenyl-2-isoxazoline and 1-substituted 3,5-diphenyl-2-pyrazolines)

Earlier tentatively assigned structures fortrans-chalcone oxime, semicarbazone and thiosemicarbazone, resp., have been corrected by means of NMR spectra: open-chain structures have been found for the title compounds formed in the acid catalyzed reaction, whereas basic catalysts lead to intramolecular cyclization with formation of 3,5-diphenyl-2-isoxazoline and 1-substituted 3,5-diphenyl-2-pyrazolines resp. Assignments in the NMR spectra were made by comparison with those from authentic samples oftrans-dypnone and its semicarbazone.

     

Topological/chemical distances and graph centers in molecular graphs with multiple bonds

A set of priority rules is defined for obtaining the graph center in multigraphs, i.e. in molecular graphs with multiple bonds. In order to obtain closer agreement between experimental bond distances in molecules and the relative magnitudes of entries in the distance matrix, a new type of distance is defined, called the “chemical distance”: CD = b^t-14, where b is the conventional bond multiplicity.     

Janus-type isopropyl groups revealed by 1H-NMR spectra of alkylpyridines with lanthanide shift reagents (LSR)

Two Isomeric pyridines were designed and prepared : 2-isopropyl-4,5,6-trimethyl-(3) and 2-isopropyl-3,4,6-trimethyl-pyridine (7); the latter, with a buttressed isopropyl, leads to much lower induced shifts by Eu(dpm)₃ and Pr(dpm)₃ than the former, owing to the conformation in which the Janus-type iPr group how to the LSR a t-butyl-like face.     

Unique description of chemical structures based on hierarchically ordered extended connectivities (HOC procedures). III. Topological,chemical, and stereochemical coding of molecular structure

A topological code is devised on the basis of the unique topological representation of the molecule described in the preceding two parts of this series.¹ By adding to the topological code additional chemical information on atoms and/or bonds, as well as stereochemical information, a chemical and respectively stereochemical code (SHOC) are also constructed. The advantages of the new linear codes are that they are convention-free codes, preserving the symmetry of molecular graph, and easily implemented either manually or by means of computer programs. By concentrating all topological, chemical, and stereochemical information, our code (SHOC) is more compact and more general than the codes based on several separate lists.     

Graph-theoretical analysis of the bonding topology in polyhedral organic cations

It is shown that in several cases where planar delocalisation in organic cations would result in the formation of an anti-aromatic system, polyhedral delocalisation is the form of bonding actually preferred. This explains, for instance, why organic cations in such cases adopt cage-like structures. A full graph-theoretical analysis, similar to one previously published¹² for polyhedral boranes, carboranes and metal clusters, indicates that the nido structure for (CH)₅⁺ may readily be accounted for. Moreover, in the case of the dication (CH)₆²⁺ the fact that its energy minimum also corresponds to a nido structure is explained. In fact, no closo- or arachno-type structures appear to be possible for organic cations. A number of structural predictions concerning these species are given in the conclusion.