Zhang–Zhang Polynomials of Ribbons

He, Bing; Chou, Chien Pin; Langner, Johanna; Witek, Henryk A.

doi:10.3390/sym12122060

Cited by 8 publications

(5 citation statements)

References 25 publications

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“…The remaining diagonal element

corresponds to the

path area, which has a shape of an elementary benzenoid, a ribbon

, with the ZZ polynomial given by

. (for details, see Equation (10) in [ 35 ]). Extensive numerical experimentation allows to establish that the value

located on the

superdiagonal in the last column of

with odd m can be expressed by

where

, with

and

given by Equations ( 22 ) or ( 23 ).…”

Section: Discovery Of the Determinantal Formulasmentioning

confidence: 99%

“…The most convenient way of representing the subsequence

is given in the form of its generating function

which is most often referred to as the Clar covering polynomial or, from the names of its inventors, as the Zhang–Zhang polynomial or the ZZ polynomial of

[ 9 , 10 , 11 , 12 , 13 , 14 , 15 ]. Substantial research effort has been invested in the determination of

for elementary families of benzenoids [ 8 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 ]. The rapid development of Clar theory stimulated by these discoveries in recent years has led to many new interesting applications and connections to other branches of chemistry, graph theory, and combinatorics [ 8 , 17 , 18 , 19 , 21 , 28 ,…”

Section: Introductionmentioning

confidence: 99%

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Zhang–Zhang Polynomials of Multiple Zigzag Chains Revisited: A Connection with the John–Sachs Theorem

Witek

2021

Molecules

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Multiple zigzag chains Zm,n of length n and width m constitute an important class of regular graphene flakes of rectangular shape. The physical and chemical properties of these basic pericondensed benzenoids can be related to their various topological invariants, conveniently encoded as the coefficients of a combinatorial polynomial, usually referred to as the ZZ polynomial of multiple zigzag chains Zm,n. The current study reports a novel method for determination of these ZZ polynomials based on a hypothesized extension to John–Sachs theorem, used previously to enumerate Kekulé structures of various benzenoid hydrocarbons. We show that the ZZ polynomial of the Zm,n multiple zigzag chain can be conveniently expressed as a determinant of a Toeplitz (or almost Toeplitz) matrix of size m2×m2 consisting of simple hypergeometric polynomials. The presented analysis can be extended to generalized multiple zigzag chains Zkm,n, i.e., derivatives of Zm,n with a single attached polyacene chain of length k. All presented formulas are accompanied by formal proofs. The developed theoretical machinery is applied for predicting aromaticity distribution patterns in large and infinite multiple zigzag chains Zm,n and for computing the distribution of spin densities in biradical states of finite multiple zigzag chains Zm,n.

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“…The remaining diagonal element

corresponds to the

path area, which has a shape of an elementary benzenoid, a ribbon

, with the ZZ polynomial given by

. (for details, see Equation (10) in [ 35 ]). Extensive numerical experimentation allows to establish that the value

located on the

superdiagonal in the last column of

with odd m can be expressed by

where

, with

and

given by Equations ( 22 ) or ( 23 ).…”

Section: Discovery Of the Determinantal Formulasmentioning

confidence: 99%

“…The most convenient way of representing the subsequence

is given in the form of its generating function

which is most often referred to as the Clar covering polynomial or, from the names of its inventors, as the Zhang–Zhang polynomial or the ZZ polynomial of

[ 9 , 10 , 11 , 12 , 13 , 14 , 15 ]. Substantial research effort has been invested in the determination of

Section: Introductionmentioning

confidence: 99%

Zhang–Zhang Polynomials of Multiple Zigzag Chains Revisited: A Connection with the John–Sachs Theorem

Witek

2021

Molecules

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“…Completely new vistas in the Clar theory have been recently opened by the development of the interface theory of benzenoids [29,38,40,41,58]. It has been demonstrated that the description of resonance structures of a benzenoid B can be reduced to studying the covering characters of its interfaces.…”

Section: General Introductionmentioning

confidence: 99%

ZZ of regular m-tier benzenoid strips as extended strict order polynomials of associated posets part 1. Proof of equivalence

Langner¹,

Witek²

2021

match

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In Part 1 of the current series of papers, we demonstrate the equivalence between the Zhang-Zhang polynomial ZZ(S, x) of a Kekuléan regular m-tier strip S of length n and the extended strict order polynomial E • S (n, x + 1) of a certain partially ordered set (poset) S associated with S. The discovered equivalence is a consequence of the one-to-one correspondence between the set {K} of Kekulé structures of S and the set {µ : S ⊃ A → [ n ]} of strictly order-preserving maps from the induced subposets of S to the interval [ n ]. As a result, the problems of determining the Zhang-Zhang polynomial of S and of generating the complete set of Clar covers of S reduce to the problem of constructing the set L(S) of linear extensions of the corresponding poset S and studying their basic properties. In particular, the Zhang-Zhang polynomial of S can be written in a compact form as ZZ(S, x) = |S| k=0 w∈L(S)

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“…No computer program is available to date based on these concepts, but we hope that such a code—enabling one to surpass the limit of 500 carbon atoms in ZZ polynomial calculations for pericondensed benzenoids—will be made available soon. 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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Zhang–Zhang Polynomials of Ribbons

Cited by 8 publications

References 25 publications

Zhang–Zhang Polynomials of Multiple Zigzag Chains Revisited: A Connection with the John–Sachs Theorem

Zhang–Zhang Polynomials of Multiple Zigzag Chains Revisited: A Connection with the John–Sachs Theorem

ZZ of regular m-tier benzenoid strips as extended strict order polynomials of associated posets part 1. Proof of equivalence

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

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