Clar and sextet polynomials of buckminsterfullerene

Shiu, Wai Chee; Lam, Peter Che Bor; Zhang, Heping

doi:10.1016/s0166-1280(02)00649-8

Cited by 14 publications

(13 citation statements)

References 26 publications

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“…Since it has the Fries structure so that each hexagon is alternating, any set of disjoint hexagons always forms a sextet pattern. Based upon this, Shiu et al [68] computed the sextet polynomial of 60 C as 8 Ye et al [69] showed that every hexagon of a fullerene is resonant, determined all the other eight 3-resonant fullerenes (i.e. every set of at most three disjoint hexagons forms a sextet pattern) and proved that any independent hexagons of a 3-resonant fullerene graph form a sextet pattern.…”

Section: Resonant Patterns and Kekulé Structures -Non-alternant Casementioning

confidence: 99%

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Advances of Clar's Aromatic Sextet Theory and Randic´'s Conjugated Circuit Model

Zhang¹

2011

TOOCJ

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Clar's aromatic sextet theory provides a good means to describe the aromaticity of benzenoid hydrocarbons, which was mainly based on experimental observations. Clar defined sextet pattern and Clar number of benzenoid hydrocarbons, and he observed that for isomeric benzenoid hydrocarbons, when Clar number increases the absorption bands shift to shorter wavelength, and the stability of these isomers also increases. Motivated by Clar's aromatic sextet theory, three types of polynomials (sextet polynomial, Clar polynomial, and Clar covering polynomial) were defined, and Randic´'s conjugated circuit model was also established. In this survey we attempt to review some advances on Clar's aromatic sextet theory and Randic´'s conjugated circuit model in the past two decades. New applications of these polynomials to fullerenes, and calculation methods of linear independent and minimal conjugated circuit polynomials of benzenoid hydrocarbons are also presented.

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Section: Resonant Patterns and Kekulé Structures -Non-alternant Casementioning

confidence: 99%

“…Shiu et al [68] clarified such an extended Clar structure by replacing (c) with (d): the set of circles is maximal, i.e. no new cycle can be drawn using (a) and (b).…”

Section: Clar Polynomialmentioning

confidence: 99%

Advances of Clar's Aromatic Sextet Theory and Randic´'s Conjugated Circuit Model

Zhang¹

2011

TOOCJ

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“…where σ(G, i) denotes the number of sextet patterns of G with i hexagons, and C(G) the Clar number of G. The sextet polynomial of C 60 is computed [24] as B C 60 (x) = 5x 8 + 320x 7 + 1240x 6 + 1912x 5 + 1510x 4 + 660x 3 + 160x 2 + 20x + 1. (8) For a detailed discussion and review of sextet polynomials, see [14,22].…”

Section: Sextet Polynomials Of 3-resonant Fullerene Graphsmentioning

confidence: 99%

Onk-Resonant Fullerene Graphs

Ye¹,

Zhong-bin

Zhang³

2009

SIAM J. Discrete Math.

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A fullerene graph F is a 3-connected plane cubic graph with exactly 12 pentagons and the remaining hexagons. Let M be a perfect matching of F . A cycle C of F is M -alternating if the edges of C appear alternately in and off M . A set H of disjoint hexagons of F is called a resonant pattern (or sextet pattern) if F has a perfect matching M such that all hexagons in H are M -alternating. A fullerene graph F is k-resonant if any i (0 ≤ i ≤ k) disjoint hexagons of F form a resonant pattern. In this paper, we prove that every hexagon of a fullerene graph is resonant and all leapfrog fullerene graphs are 2-resonant. Further, we show that a 3-resonant fullerene graph has at most 60 vertices and construct all nine 3-resonant fullerene graphs, which are also k-resonant for every integer k > 3. Finally, sextet polynomials of the 3-resonant fullerene graphs are computed.

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“…The Clar polynomial and sextet polynomial of C 60 for counting Clar structures and sextet patterns respectively were computed in [18]. This implies $ This paper is supported by NSFC grant 10831001. that C 60 has 5 Clar formulas and Clar number 8 [5].…”

Section: Introductionmentioning

confidence: 99%

Extremal fullerene graphs with the maximum Clar number

Zhang

2009

Discrete Applied Mathematics

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A fullerene graph is a cubic 3-connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let $F_n$ be a fullerene graph with $n$ vertices. A set $\mathcal H$ of mutually disjoint hexagons of $F_n$ is a sextet pattern if $F_n$ has a perfect matching which alternates on and off each hexagon in $\mathcal H$. The maximum cardinality of sextet patterns of $F_n$ is the Clar number of $F_n$. It was shown that the Clar number is no more than $\lfloor\frac {n-12} 6\rfloor$. Many fullerenes with experimental evidence attain the upper bound, for instance, $\text{C}_{60}$ and $\text{C}_{70}$. In this paper, we characterize extremal fullerene graphs whose Clar numbers equal $\frac{n-12} 6$. By the characterization, we show that there are precisely 18 fullerene graphs with 60 vertices, including $\text{C}_{60}$, achieving the maximum Clar number 8 and we construct all these extremal fullerene graphs.Comment: 35 pages, 43 figure

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Clar and sextet polynomials of buckminsterfullerene

Cited by 14 publications

References 26 publications

Advances of Clar's Aromatic Sextet Theory and Randic´'s Conjugated Circuit Model

Advances of Clar's Aromatic Sextet Theory and Randic´'s Conjugated Circuit Model

Onk-Resonant Fullerene Graphs

Extremal fullerene graphs with the maximum Clar number

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