TY - JOUR

T1 - Zhang–zhang polynomials of multiple zigzag chains revisited

T2 - A connection with the john–sachs theorem

AU - Witek, Henryk A.

N1 - Publisher Copyright:
© 2021 by the author. Licensee MDPI, Basel, Switzerland.

PY - 2021/4

Y1 - 2021/4

N2 - Multiple zigzag chains Z(m, 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 Z(m, 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 Z(m, n) multiple zigzag chain can be conveniently expressed as a determinant of a Toeplitz (or almost Toeplitz) matrix of size ⌈ m/ 2 ⌉×⌈m/2 ⌉ consisting of simple hypergeometric polynomials. The presented analysis can be extended to generalized multiple zigzag chains Zk (m, n), i.e., derivatives of Z(m, 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 Z(m, n) and for computing the distribution of spin densities in biradical states of finite multiple zigzag chains Z(m, n).

AB - Multiple zigzag chains Z(m, 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 Z(m, 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 Z(m, n) multiple zigzag chain can be conveniently expressed as a determinant of a Toeplitz (or almost Toeplitz) matrix of size ⌈ m/ 2 ⌉×⌈m/2 ⌉ consisting of simple hypergeometric polynomials. The presented analysis can be extended to generalized multiple zigzag chains Zk (m, n), i.e., derivatives of Z(m, 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 Z(m, n) and for computing the distribution of spin densities in biradical states of finite multiple zigzag chains Z(m, n).

KW - Benzenoids

KW - Clar covering polynomials

KW - Enumeration of Clar covers

KW - John–Sachs theorem

KW - Multiple zigzag chains

KW - ZZ polynomials

UR - http://www.scopus.com/inward/record.url?scp=85105225722&partnerID=8YFLogxK

U2 - 10.3390/molecules26092524

DO - 10.3390/molecules26092524

M3 - Article

C2 - 33925975

AN - SCOPUS:85105225722

SN - 1420-3049

VL - 26

JO - Molecules

JF - Molecules

IS - 9

M1 - 2524

ER -