The Clar formulas of a benzenoid system and the resonance graph

Salem, Khaled; Klavar, Sandi; Vesel, Aleksander; igert, Petra

doi:10.1016/j.dam.2009.02.016

Cited by 13 publications

(13 citation statements)

References 21 publications

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“…The original version of the Clar method is qualitative and has no direct foundation in quantum theory. Eventually, much effort was devoted to providing a quantitative and theoretically founded re-formulation of the Clar model (see the recent works [18][19][20][21][22][23][24][25][26] and the references cited therein). In earlier studies, 13,14,27,28 examples of benzenoid hydrocarbons were found in which the predictions of Clar theory were violated.…”

Section: Introductionmentioning

confidence: 99%

A test of Clar aromatic sextet theory

Gutman

Radenković

Antić

et al. 2013

JSCS

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The Clar aromatic sextet theory predicts that the intensity of cyclic conjugation in chevron-type benzenoid hydrocarbons monotonically decreases along the central chain. This regularity has been tested by means of several independent theoretical methods (by the energy effects of the respective sixmembered rings, as well as by their HOMA, NICS, and SCI values, calculated at the B3LYP/6-311G(d,p) level of DFT theory). Our results show that the predictions of Clar theory are correct only for the first few members of the chevron homologous series, and are violated at the higher members. This indicates that Clar theory is not universally applicable, even in the case of fully conjugated benzenoid molecules.

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Section: Introductionmentioning

confidence: 99%

A test of Clar aromatic sextet theory

Gutman

Radenković

Antić

et al. 2013

JSCS

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“…It was proved in [15] that the resonance graphs of a special class of benzenoid graphs, fibonacenes, are precisely Fibonacci cubes. On the other hand, as proved in [21], the Clar number Cl(B) of a benzenoid system B is equal to the number of subgraphs of the resonance graph of B isomorphic to Q Cl(B) . Hence enumeration of hypercubes of all dimensions in Fibonacci cubes is essential there.…”

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confidence: 99%

Cube Polynomial of Fibonacci and Lucas Cubes

Klavžar

Mollard

2011

Acta Appl Math

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The cube polynomial of a graph is the counting polynomial for the number of induced k-dimensional hypercubes (k ≥ 0). We determine the cube polynomial of Fibonacci cubes and Lucas cubes, as well as the generating functions for the sequences of these cubes. Several explicit formulas for the coefficients of these polynomials are obtained, in particular they can be expressed with convolved Fibonacci numbers. Zeros of the studied cube polynomials are explicitly determined. Consequently, the coefficients sequences of cube polynomials of Fibonacci and Lucas cubes are unimodal.

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“…Characterization and enumeration of Clar covers [1][2][3][4][5][6][7][8] of an arbitrary benzenoid containing up to about 500 carbon atoms became almost a routine task after introducing the concept of the Clar covering polynomial (aka Zhang-Zhang polynomial or briefly ZZ polynomial) [9][10][11][12][13] and development of ZZDecomposer, 14,15 an automatized computer program using recursive properties of ZZ polynomials to compute them in a robust manner. 4,9,[16][17][18] ZZDecomposer is freely available 19,20 and can be used for a convenient graphical definition of an arbitrary benzenoid by mouse selection of desired hexagonal faces or edges of the underlying hexagonal honeycomb lattice followed by brute-force recursive computation of the ZZ polynomial, finding recursive ZZ polynomial decomposition pathways, and producing camera-ready images of such pathways.…”

Section: Introductionmentioning

confidence: 99%

Hexagonal flakes as fused parallelograms: A determinantal formula for Zhang‐Zhang polynomials of the O(2, m, n) benzenoids

Langner

Witek

2021

J Chinese Chemical Soc

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We report a determinantal formula for the Zhang-Zhang polynomial of the hexagonal flake O(2, m, n) applicable to arbitrary values of the structural parameters m and n. The reported equation has been discovered by extensive numerical experimentation and is given here without a proof. Our combinatorial analysis performed on a large collection of isostructural O(2, m, n) benzenoids yielded a ZZ polynomial formula corresponding to the determinant of a certain 2 × 2 matrix referred to by us as the generalized John-Sachs path matrix, because of the striking structural similarity with the original path matrices introduced by the John-Sachs theory of Kekulé structures. The presented conjecture hints at the existence of a generalization of the John-Sachs theory applicable to characterization and enumeration of Clar covers.

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The Clar formulas of a benzenoid system and the resonance graph

Abstract: a b s t r a c tIt is shown that the number of Clar formulas of a Kekuléan benzenoid system B is equal to the number of subgraphs of the resonance graph of B isomorphic to the Cl(B)-dimensional hypercube, where Cl(B) is the Clar number of B.

Cited by 13 publications

References 21 publications

A test of Clar aromatic sextet theory

A test of Clar aromatic sextet theory

Cube Polynomial of Fibonacci and Lucas Cubes

Hexagonal flakes as fused parallelograms: A determinantal formula for Zhang‐Zhang polynomials of the O(2, m, n) benzenoids

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