Browsing by Author "Langner, Johanna"

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Algorithm for generating generalized resonance structures of single zigzag chains based on interface theory
(2018-05-01)
Langner, Johanna
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Witek, Henryk A.
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應用化學系
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應用化學系分子科學碩博班
;
Department of Applied Chemistry
An alternative formulation of the theory of generalized resonance structures (Clar covers) of single zigzag chains N(n), based on the novel concept of interfaces, is presented. The global structure of every Clar cover can be conveniently expressed as a unique sequence of n interfaces. The complete set of Clar covers is then robustly constructed as the complete set of walks of length on the associated interface connectivity graph. The presented algorithm is readily generalizable to an arbitrary class of benzenoid structures.
Connectivity Graphs for Single Zigzag Chains and their Application for Computing ZZ Polynomials
(2017-12-01)
Langner, Johanna
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Witek, Henryk A.
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應用化學系
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應用化學系分子科學碩博班
;
Department of Applied Chemistry
We present a graph-theoretical interpretation of the recently developed interface theory of single zigzag chains N(n) in form of connectivity graphs. A remarkable property of the connectivity graphs is the possibility of constructing the full set of Clar covers of N(n) as the complete set of walks on these graphs. Connectivity graphs can be constructed in a direct form, in which the number of vertices is growing linearly with the length n of N(n), and in a reduced form, in which the number of vertices is independent of the actual length of the chain. The presented results can be immediately used for the determination of the Zhang-Zhang (ZZ) polynomials of N(n) in an easy and natural manner. Two techniques for computing the ZZ polynomial are proposed, one based on direct recursive computations and the other based on a general solution to a set of recurrence relations. Generalization of the interface theory to arbitrary benzenoid structures, the construction of associated connectivity graphs, and techniques for the computation of the associated ZZ polynomials will be presented in the near future.
Equivalence between Clar covering polynomials of single zigzag chains and tiling polynomials of 2 x n rectangles
(2018-07-10)
Langner, Johanna
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Witek, Henryk A.
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應用化學系
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應用化學系分子科學碩博班
;
Department of Applied Chemistry
This paper offers a formal explanation of a rather puzzling and surprising equivalence between the Clar covering polynomials of single zigzag chains and the tiling polynomials of 2 x n rectangles for tilings using 1 x 2, 2 x 1 and 2 x 2 tiles. It is demonstrated that the set of Clar covers of single zigzag chains N(n - 1) is isomorphic to the set of filings of a 2 x n rectangle. In particular, this isomorphism maps Clar covers of N(n - 1) with k aromatic sextets to tilings of a 2 x n rectangle using k square 2 x 2 tiles. The proof of this fact is an application of the recently introduced interface theory of Clar covers. The existence of a similar relationship between the Clar covers of more general benzenoid structures and more general tilings of rectangles remains an interesting open problem in chemical graph theory. (C) 2018 Elsevier B.V. All rights reserved.
Interface Theory of Benzenoids
(2020-01-01)
Langner, Johanna
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Witek, Henryk A.
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交大名義發表
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應用化學系分子科學碩博班
;
National Chiao Tung University
We propose here a novel tool for chemical graph theory, referred to in the following as the interface theory of benzenoids, designed for constructing, enumerating, and characterizing Clar covers of arbitrary benzenoids. The presented interface theory uses the new concepts of fragments and interfaces to express an arbitrary Clar cover of a benzenoid B as a collection of covering characters for the interface bonds in B. A set of fundamental theorems demonstrates that the interface theory is internally consistent. In particular, we are able to establish the necessary and sufficient conditions required for the existence and uniqueness of a Clar cover in terms of interface bond coverings. The proposed theoretical framework can be conveniently and efficiently used for determining whether a given benzenoid is Kekulean, for finding its Clar covers and Kekule structures, for computing its ZZ polynomials, as well as deriving closed forms of ZZ polynomials for whole families of structurally related benzenoids, and for many other interesting applications. We believe that the presented here interface theory of benzenoids opens up new vistas in chemical graph theory and may revolutionize the field in the future.
Interface Theory of Benzenoids: Basic Applications
(2020-01-01)
Langner, Johanna
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Witek, Henryk A.
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交大名義發表
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應用化學系分子科學碩博班
;
National Chiao Tung University
We show that the recently introduced interface theory of benzenoids [Langner and Witek, MATCH Commun. Math. Comput. Chem. (2020)] can be readily applied to characterize various properties of benzenoids and their Clar covers. A carefully selected collection of examples demonstrates how to use interface theory for detecting non-Kekulean or essentially disconnected character of benzenoids, computing their Clar numbers, constructing and enumerating their Kekule structures and Clar covers, and determining their Zhang-Zhang polynomials. The exposition has rather pedagogical character and is supposed to familiarize the reader with the basic toolkit of the interface theory of benzenoids.
Zhang-Zhang Polynomials of Multiple Zigzag Chains
(2018-01-01)
Langner, Johanna
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Witek, Henryk A.
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Mos, Grzegorz
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應用化學系
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應用化學系分子科學碩博班
Generating functions of the Zhang-Zhang polynomials of multiple zigzag chains Z(m, n) and generalized multiple zigzag chains Z(k)(m, n) are derived for arbitrary values of the indices. These generating functions can be expressed in the form of highly regular finite continued fractions, Sigma(infinity)(m=0) ZZ(Z(m, n),z)t(m) = [0; -t,(-1)(2)zt,(-1)(3)zt,&,(-1)(n)zt, 1 + (-1)(n+1) zt], or, in the case of Z(k)(m, n), products of such continued fractions. For the particularly important case of the multiple zigzag chains Z(m, n), the generating functions are expanded to yield a closed form for the Zhang-Zhang polynomials of multiple zigzag chains Z(m, n) that is valid for arbitrary values of m and n.
Zhang-Zhang Polynomials of Regular 5-tier Benzenoid Strips
(2017-01-01)
Witek, Henryk A.
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Langner, Johanna
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Mos, Grzegorz
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Chou, Chien-Pin
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應用化學系
Formal derivations of closed-form expressions for Clar covering polynomials (aka Zhang Zhang polynomials or ZZ polynomials) of twelve classes of regular 5-tier benzenoid strips are presented. The derived formulas, together with the results reported previously in the literature, complete the full collection of ZZ polynomial formulas for the regular 5-tier benzenoid strips.