Closed-form formulas for the Zhang-Zhang polynomials of benzenoid structures

Prolate rectangles and their generalizations

Chien Pin Chou, Jin Su Kang, Henryk A. Witek

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

We show that the Zhang-Zhang (ZZ) polynomial of a benzenoid obtained by fusing a parallelogram M(m,n) with an arbitrary benzenoid structure ABC can be simply computed as a product of the ZZ polynomials of both fragments. It seems possible to extend this important result also to cases where both fused structures are arbitrary Kekuléan benzenoids. Formal proofs of explicit forms of the ZZ polynomials for prolate rectangles Pr(m,n) and generalized prolate rectangles Pr([m1,m2,. .,mn],n) follow as a straightforward application of the general theory, giving ZZ(Pr(m,n),x)=(1+(1+x){dot operator}m)n and ZZ(Pr([m1,m2,. .,mn],n),x)=∏k=1n(1+(1+x){dot operator}mk).

Original languageEnglish
JournalDiscrete Applied Mathematics
DOIs
Publication statusAccepted/In press - 2014 Sep 18
Externally publishedYes

Fingerprint

Rectangle
Closed-form
Polynomials
Polynomial
Parallelogram
Formal Proof
Arbitrary
Operator
Fragment
Generalization
Form

Keywords

  • Clar cover
  • Clar structure
  • Perfect matching
  • Zhang-Zhang polynomial

ASJC Scopus subject areas

  • Applied Mathematics
  • Discrete Mathematics and Combinatorics

Cite this

@article{5afd0272d02c4e5b8acf3a65b9f9cd8a,
title = "Closed-form formulas for the Zhang-Zhang polynomials of benzenoid structures: Prolate rectangles and their generalizations",
abstract = "We show that the Zhang-Zhang (ZZ) polynomial of a benzenoid obtained by fusing a parallelogram M(m,n) with an arbitrary benzenoid structure ABC can be simply computed as a product of the ZZ polynomials of both fragments. It seems possible to extend this important result also to cases where both fused structures are arbitrary Kekul{\'e}an benzenoids. Formal proofs of explicit forms of the ZZ polynomials for prolate rectangles Pr(m,n) and generalized prolate rectangles Pr([m1,m2,. .,mn],n) follow as a straightforward application of the general theory, giving ZZ(Pr(m,n),x)=(1+(1+x){dot operator}m)n and ZZ(Pr([m1,m2,. .,mn],n),x)=∏k=1n(1+(1+x){dot operator}mk).",
keywords = "Clar cover, Clar structure, Perfect matching, Zhang-Zhang polynomial",
author = "Chou, {Chien Pin} and Kang, {Jin Su} and Witek, {Henryk A.}",
year = "2014",
month = "9",
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doi = "10.1016/j.dam.2015.06.020",
language = "English",
journal = "Discrete Applied Mathematics",
issn = "0166-218X",
publisher = "Elsevier",

}

TY - JOUR

T1 - Closed-form formulas for the Zhang-Zhang polynomials of benzenoid structures

T2 - Prolate rectangles and their generalizations

AU - Chou, Chien Pin

AU - Kang, Jin Su

AU - Witek, Henryk A.

PY - 2014/9/18

Y1 - 2014/9/18

N2 - We show that the Zhang-Zhang (ZZ) polynomial of a benzenoid obtained by fusing a parallelogram M(m,n) with an arbitrary benzenoid structure ABC can be simply computed as a product of the ZZ polynomials of both fragments. It seems possible to extend this important result also to cases where both fused structures are arbitrary Kekuléan benzenoids. Formal proofs of explicit forms of the ZZ polynomials for prolate rectangles Pr(m,n) and generalized prolate rectangles Pr([m1,m2,. .,mn],n) follow as a straightforward application of the general theory, giving ZZ(Pr(m,n),x)=(1+(1+x){dot operator}m)n and ZZ(Pr([m1,m2,. .,mn],n),x)=∏k=1n(1+(1+x){dot operator}mk).

AB - We show that the Zhang-Zhang (ZZ) polynomial of a benzenoid obtained by fusing a parallelogram M(m,n) with an arbitrary benzenoid structure ABC can be simply computed as a product of the ZZ polynomials of both fragments. It seems possible to extend this important result also to cases where both fused structures are arbitrary Kekuléan benzenoids. Formal proofs of explicit forms of the ZZ polynomials for prolate rectangles Pr(m,n) and generalized prolate rectangles Pr([m1,m2,. .,mn],n) follow as a straightforward application of the general theory, giving ZZ(Pr(m,n),x)=(1+(1+x){dot operator}m)n and ZZ(Pr([m1,m2,. .,mn],n),x)=∏k=1n(1+(1+x){dot operator}mk).

KW - Clar cover

KW - Clar structure

KW - Perfect matching

KW - Zhang-Zhang polynomial

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