scholarly journals Fullerenes with the maximum Clar number

2016 ◽  
Vol 202 ◽  
pp. 58-69 ◽  
Author(s):  
Yang Gao ◽  
Qiuli Li ◽  
Heping Zhang
Keyword(s):  
Clar Number

On the Clar number of graphene fragment

2021 ◽  
Vol 59 (2) ◽  
pp. 542-553
Author(s):  
Ye Tian ◽  
Biao Zhao
Keyword(s):  
Clar Number

An Upper Bound for the Clar Number of Fullerene Graphs

2006 ◽  
Vol 41 (2) ◽  
pp. 123-133 ◽  
Author(s):  
Heping Zhang ◽  
Dong Ye
Keyword(s):  
Upper Bound ◽  
Clar Number ◽  
Fullerene Graphs

scholarly journals Unimodularity of the Clar number problem

2007 ◽  
Vol 420 (2-3) ◽  
pp. 441-448 ◽  
Author(s):  
Hernán Abeledo ◽  
Gary W. Atkinson
Keyword(s):  
Clar Number ◽  
Number Problem

scholarly journals Zhang–Zhang Polynomials of Ribbons

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2060
Author(s):  
Bing-Hau He ◽  
Chien-Pin Chou ◽  
Johanna Langner ◽  
Henryk A. Witek
Keyword(s):  
Closed Form ◽  
Topological Invariants ◽  
Important Class ◽  
Kekulé Structures ◽  
Clar Number ◽  
Interface Theory ◽  
Clar Structures ◽  
Closed Form Formula ◽  
Clar Covering Polynomial

We report a closed-form formula for the Zhang–Zhang polynomial (also known as ZZ polynomial or Clar covering polynomial) of an important class of elementary peri-condensed benzenoids Rbn1,n2,m1,m2, usually referred to as ribbons. A straightforward derivation is based on the recently developed interface theory of benzenoids [Langner and Witek, MATCH Commun. Math. Comput. Chem.2020, 84, 143–176]. The discovered formula provides compact expressions for various topological invariants of Rbn1,n2,m1,m2: the number of Kekulé structures, the number of Clar covers, its Clar number, and the number of Clar structures. The last two classes of elementary benzenoids, for which closed-form ZZ polynomial formulas remain to be found, are hexagonal flakes Ok,m,n and oblate rectangles Orm,n.

scholarly journals Clar Structure and Fries Set of Fullerenes and (4,6)-Fullerenes on Surfaces

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Yang Gao ◽  
Heping Zhang
Keyword(s):  
Projective Plane ◽  
Euler Characteristic ◽  
Clar Number ◽  
Clar Structure ◽  
Fries Number

Fowler and Pisanski showed that the Fries number for a fullerene on surface Σ is bounded above by|V|/3, and fullerenes which attain this bound are exactly the class of leapfrog fullerenes on surface Σ. We showed that the Clar number of a fullerene on surface Σ is bounded above by(|V|/6)-χ(Σ), whereχ(Σ)stands for the Euler characteristic of Σ. By establishing a relation between the extremal fullerenes and the extremal (4,6)-fullerenes on the sphere, Hartung characterized the fullerenes on the sphereS0for which Clar numbers attain(|V|/6)-χ(S0). We prove that, for a (4,6)-fullerene on surface Σ, its Clar number is bounded above by(|V|/6)+χ(Σ)and its Fries number is bounded above by(|V|/3)+χ(Σ), and we characterize the (4,6)-fullerenes on surface Σ attaining these two bounds in terms of perfect Clar structure. Moreover, we characterize the fullerenes on the projective planeN1for which Clar numbers attain(|V|/6)-χ(N1)in Hartung’s method.

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