Topological properties of benzenoid systems. XXXV. Number ofKekulé structures of multiple-chain aromatics

Recurrence relations and explicit combinatorial expressions are derived for the number ofKekulé structures of certain multiple-chain condensed aromatics.

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Topologische Eigenschaften benzenoider Systeme, 35. Mitt.: Anzahl vonKekulé-Strukturen von Mehrfachketten-Aromaten

Zusammenfassung Es werden Beziehungen und explizite kombinatorische Ausdrücke für die Anzahl möglicherKekulé-Strukturen bestimmter mehrfachkettiger kondensierter Aromaten abgeleitet.

     

Contribution to the techniques of enumeration ofKekulé structures

Four auxiliary classes of benzenoids are introduced, and formulas are given for their number ofKekulé structures (K). An enumeration method forK of different important classes of benzenoids is illustrated by examples. The utilization of essentially disconnected benzenoids is a special feature of the method.

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Ein Beitrag zur Methodik der Bestimmung möglicherKekulé-Strukturen

Zusammenfassung Es werden Hilfsklassen von benzenoiden Aromaten eingeführt und Formeln zur zahlenmäßigen Auswertung möglicherKekulé-Strukturen angegeben (K). Die Auswertungsmethode fürK wird an verschiedenen wichtigen Klassen benzenoider Verbindungen exemplarisch gezeigt. Die Verwendbarkeit essentiell abgekoppelter aromatischer Bauelemente ist eine spezielle Eigenschaft dieser Methode.

     

Enumeration ofKekulé structures:Étagères and related benzenoid classes

Formulas are developed for the number ofKekulé structures of some benzenoid classes, which are interpreted in terms of annelations of a multiple linear chain (parallelogram). One-sided and two-sided annelations are considered, the latter category leading to the definition of a benzenoid class referred to asétagères.

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Ermittlung der Anzahl vonKekulé-Strukturen: Étagère und verwandte benzenoide Klassen

Zusammenfassung Es werden Formeln für die Ermittlung der Anzahl möglicherKekulé-Strukturen für einige benzenoiden Klassen angegeben, wobei diese durch annelierte, mehrfache, lineare Ketten (Parallelogramme) charakterisiert werden. Dabei wird einseitige und zweiseitige Annelierung berücksichtigt, für die letztgenannte Gruppe wird der Ausdruck Étagère geprägt.

     

Normal Components, Kekulé Patterns, and Clar Patterns in Plane Bipartite Graphs

As a general case of molecular graphs of polycyclic alternant hydrocarbons, we consider a plane bipartite graph G with a Kekulé pattern (perfect matching). An edge of G is called nonfixed if it belongs to some, but not all, perfect matchings of G. Several criteria in terms of resonant cells for determining whether G is elementary (i.e., without fixed edges) are reviewed. By applying perfect matching theory developed in plane bipartite graphs, in a unified and simpler way we study the decomposition of plane bipartite graphs with fixed edges into normal components, which is shown useful for resonance theory, in particular, cell and sextet polynomials. Further correspondence between the Kekulé patterns and Clar (resonant) patterns are revealed.     

Topological properties of benzenoid systems

The sextet polynomial of non-branched cata-condensed benzenoid molecules is proved to be related (Eq. (3)) to the characteristic polynomial of a tree.     

Kekulé structure counts: General formulations for primitive coronoid hydrocarbons

Summary The Kekulé structure counts (K) for primitive coronoids are treated. TheK formula which involves the trace of a matrix product is recalled and supplemented with new findings. In this way a kind of symmetry in the mathematical formulations is restored. Another general formulation for theK number is provided in terms of polynomials which, for a somewhat mysterious reason, are identified as the matching polynomials of cycles.

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Zählung von Kekulé-Strukturen

Zusammenfassung Es werden Kekulé-Strukturzählungen (K) für einfache Coronoide behandelt. Die Formel fürK wird durch neugefundene Eigenschaften ergänzt. So wird eine Art von Symmetrie in den mathematischen Formulierungen erreicht. Eine andere generelle Formulierung derK-Zahlen wird in Form von Polynomen bereitgestellt, welche aus uneinsichtigen Gründen als die passenden Polynome von Cyclen identifiziert wurden.

     

Topological properties of benzenoid systems. Bounds and approximate formulae for total π-electron energy

A novel variational approach to McClelland’s estimate of total π-electron energy (E) is described. An extension of this method yields lower and upper bounds andE of benzenoid hydrocarbons. On the basis of these bounds several approximate topological formulae forE are obtained.     

Balancing the Pauling bond orders in Kekuléan benzenoid hydrocarbons

We consider a cutting of the molecular graph B of a Kekuléan benzenoid molecule into two disconnected subgraphs, S and the other, by deleting from B certain edges. It is required that both subgraphs remain Kekuléan. The edges involved in this cutting are classified as starred and unstarred. A starred edge is incident to a starred carbon site of the subgraph S, whereas an unstarred edge to an unstarred carbon site of S. The following regularity is established: for any above-described cutting of any Kekuléan benzenoid system, the sum of the Pauling bond orders of the starred edges is equal to that for the unstarred edges.     

Note on the application of the reduced graph model in conjunction with search trees to the enumeration ofKekulé structures

The reduced graph model, when used in conjunction with the search trees method, provides a novel combinatorial procedure for the enumeration and generation ofKekulé structures. The procedure is suited for large benzenoid hydrocarbons consisting of cata- and thin peri-condensed parts.

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Zur Anwendung des Reduced Graph Model im Zusammenhang mit Search Trees zur Ermittlung der Anzahl möglicherKekulé-Strukturen

Zusammenfassung Das Modell erlaubt mit der im Titel genannten Kombination eine neuartige Methode zur Ermittlung und Generierung vonKekulé-Strukturen. Das Verfahren ist für große benzoide Kohlenwasserstoffe geeignet, die aus cata- und (dünnen) peri-kondensierten Teilstrukturen bestehen.

     

The Gutman Formulas for Algebraic Structure Count

The concept of ASC (algebraic structure count) is introduced into theoretical organic chemistry by Wilcox as the difference between the number of so-called even and odd Kekulé structures of a conjugated molecule. Precisely, algebraic structure count (ASC-value) of the bipartite graph G corresponding to the skeleton of a conjugated hydrocarbon is defined by where A is the adjacency matrix of G. In the case of bipartite planar graphs containing only circuits of the length of the form 4s+2 (s=1,2,...) (the case of benzenoid hydrocarbons), this number is equal to the number of the perfect matchings (K-value) of G. However, if some of circuits are of the length 4s (s=1,2,...) then the problem of evaluation ASC-value becomes more complicated. The theorem formulated and proved in this paper gives a simple and efficient algorithm for calculation of algebraic structure count of an arbitrary bipartite graphs with n+n vertices. Three recurrence formulas for the algebraic structure count – the Gutman formulas, which are closely analogous to the well-known recurrence formula K{G}=K{G–e}+K{G–(e)} for the number of perfect matchings (G–e is the subgraph obtained from the graph G by deleting the edge e and G–(e) is the subgraph obtained from G by deleting both the edge e and its terminal vertices) are obtained as a simple corollary of the theorem.     

The spectral radius of the adjacency matrix of benzenoid hydrocarbon

The variational treatment of the spectral radius of the adjacency matrix, based on the Rayleigh's quotient and the steepest descent method is presented. The resulting formula, valid for the benzenoid hydrocarbons, gives accurate values for benzene and the linearly condensed benzenoid system of infinite length.     

On the number of Kekulé structures in capped zigzag nanotubes

Two theoretical formulae for the number of Kekulé structures in general capped zigzag nanotubes are established: one of which is by using the techniques of the transfer matrices, the other involves the eigenvalues of the transfer matrix which reveals the asymptotic behaviour of this index. In effective, according to the symmetric aspect of the tubule, the order of the transfer matrix could be notably decreased. As an application, the closed expressions for four types are given out and the relevant numerical results for those of length up to 50 are listed.     

Theory of helicenic hydrocarbons. Part 6: Concealed non-kekuléans

As a contribution to the enumeration of concealed non-Kekuléan helicenes, a construction method is applied to the caseh=14, whereh is the number of hexagons; computerized enumerations are given forh=14 and 15.     

Combinatorial library synthesis of two- and three-ring benzenoid amides on solid support

A liquid crystalline library of 100 molecules with three benzene rings and 18 molecules with two benzene rings was synthesized on solid support by means of benzoylation or acylation and palladium(0)-catalyzed carbonylation of a secondary amine obtained by a reductive amination of 4-iodoaniline and a backbone amide linker. The conversion and purity of the final products are high enough to investigate the mesomorphic properties.     

Contributions to the theory of benzenoid isomers: Some properties of extremal benzenoids

Summary An extremal benzenoid, A, is defined by having the maximum number of internal vertices for the given number of hexagons. A number of properties of A are treated, where the modes of hexagons play an important role. The processes of circumscribing and excising for benzenoids are defined and treated in detail, supported by analytical expressions. It is concluded by stating that A can always be circumscribed an unlimited number of times.

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Beiträge zur Theorie von benzenoiden Isomeren. Einige Eigenschaften extremer Benzenoide

Zusammenfassung Ein extremes Benzenoid A wird durch die maximale Anzahl von internen Vertices für eine gegebene Anzahl von Sechsecken definiert. Es werden einige Eigenschaften von A behandelt, wobei die Erscheinungsformen der Sechsecke eine wichtige Rolle spielen. Der Prozess der Umschreibung und Ausschneidung der Benzenoide wird definiert und im Detail behandelt, wobei analytische Ausdrücke zur Unterstützung herangezogen werden. Es wird geschlossen, daß A stets unbegrenzt oft umschrieben werden kann.

     

P-V matrix and enumeration of Kekulé structures

In this paper, a simple and intuitive proof of the theorem K= ¦det N(G)¦ [1,2] is given.     

Kekulé structure counts for some classes of all-benzenoid and all-coronoid hydrocarbons

Summary A general formula for the Kekulé structure count (K) is deduced for the class of catacondensed all-benzenoids with unbranched backbones. The formula is extended to thin pericondensed all-benzenoids, where allowance is made for pyrene units. In this treatment the fragmentation matrices are employed. A generalization of these matrices is furnished. Next some generalK formulas for classes of catacondensed and thin pericondensed all-coronoids are deduced. Again the fragmentation matrices are employed, but the problem is also studied in terms of certain polynomials.Dedicated to the memory of Professor Oskar E. Polansky, who died in January 1989. He was the one who coined the term all-benzenoid.     

The number ofKekulé structures of hexagon-shaped benzenoids and members of other related classes

Some results from the enumeration ofKekulé structures are reviewed; they pertain to parallelogram-shaped, bent strips, chevrons and symmetrical hexagon-shaped benzenoids. The existing formulas are extended to the class of asymmetrical hexagons. Applications of the new formula reproduces a number of known results for three-tier and four-tier strips. In the latter case also some new formulas are achieved.

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Die Anzahl derKekulé-Strukturen von Benzenoiden mit sechseckigem Umri und Mitgliedern anderer verwandter Klassen

Zusammenfassung Es wird ein Überblick über die Berechnung der Anzahl möglicherKekulé-Strukturen von benzenoiden Verbindungen unterschiedlicher Formenklassen gegeben. Dabei werden die existierenden Formeln für die Klasse asymmetrischer Sechseckformen ausgeweitet und die neue Formel auch an bekannten Ergebnissen erprobt.

     

Characterization of distribution of pi-electrons amongst benzenoid rings for Randić’s “algebraic” Kekulé structures

For benzenoid hydrocarbons the distribution of pi-electrons amongst rings is characterized in the context of Randis mode of assignment attending to the different Kekulé structures. In particular the mean and mean deviation from the mean are considered, and the benzenoids which achieve maximum deviation are identified.     

Bond length features of linear carbon chains of finite to infinite size: Visual interpretation from Pauling bond orders

Schemes for Kekulé structure counting of linear carbon chains are suggested. Mathematical formulas, which calculate the Pauling bond order P(k, N) of a chemical bond numbered by k, are given for the carbon chain with N carbon atoms. By use of the least‐squares fitting of a linearity, relationships between Pauling bond orders and bond lengths are obtained, and such correlation of the Pauling bond order–bond length can be qualitatively extended to the excited states. The relative magnitudes of Pauling bond orders in unsaturated carbon chains dominate C–C bond lengths a well as the bond length feature with the chain size increasing. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 144–149, 2003