Clar number of hexagonal chains

Salem, Khaled; Gutman, Iván

doi:10.1016/j.cplett.2004.07.031

Cited by 12 publications

(7 citation statements)

References 6 publications

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“…In addition to the obvious brute‐force calculations, ZZDecomposer has been used in numerous applications 15–17,26–37 to find closed‐form formulas of ZZ polynomials for various families of basic catacondensed and pericondensed benzenoids, substantially extending the total body of previously available results 4,9–13,38–43 . The rapid development of the Clar theory stimulated by these discoveries in recent years leads to many new interesting applications and connections to other branches of chemistry, graph theory, and combinatorics 7,8,18,31,41–63 …”

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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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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“…In particular, Abeledo and Atkinson [5,9] showed that the Clar problem could be solved in polynomial time as a linear program. Polynomial combinatorial algorithms to solve the Clar problem for special classes of hexagonal systems were studied in [9,[13][14][15]. However, it is still an open problem to design a polynomial combinatorial algorithm to solve the Clar problem in the general case.…”

Section: Introductionmentioning

confidence: 99%

Hypergraphs and the Clar problem in hexagonal systems

Salem

Abeledo

2006

Discrete Optimization

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We show that for a hypergraph H that is separable and has the Helly property, the perfect matchings of H are the strongly stable sets of the line graph of H . Also, we show that the hypergraph generated by a hexagonal system is separable and has the Helly property. Finally, we note that the Clar problem of a hexagonal system is a minimum cardinality perfect matching problem of the hypergraph generated by the hexagonal system. Hence, the Clar problem of a hexagonal system is a minimum cardinality strongly stable set problem in the line graph of the hypergraph generated by the hexagonal system.

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“…In the case of large benzenoid systems determining the value of Cl may be a rather difficult task and special methods for its calculation were developed (8)(9)(10)(11)(12).…”

Section: Introductionmentioning

confidence: 99%

“…Setting x = 0 into Eqn. (12) we obtain the (previously known (2, 32) formula K (Q n ) = 2 · 3 n . An easy analysis shows that ζ (Q n , x) is a polynomial of degree n +1, implying that the Clar number of Q n is equal to n + 1.…”

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confidence: 98%