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Leapfrog Transformation and polyhedra of Clar Type
 J. Chem. Soc. Faraday Trans
, 1994
"... The socalled leapfrog transformation that was first introduced for fullerenes (trivalent polyhedra with 12 pentagonal faces and all other faces hexagonal) is generalised to general polyhedra and maps on surfaces. All spherical polyhedra can be classified according to their leapfrog order. A polyh ..."
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The socalled leapfrog transformation that was first introduced for fullerenes (trivalent polyhedra with 12 pentagonal faces and all other faces hexagonal) is generalised to general polyhedra and maps on surfaces. All spherical polyhedra can be classified according to their leapfrog order. A polyhedron is said to be of Clar type if there exists a set of faces that cover each vertex exactly once. It is shown that a fullerene is of Clar type if and only if it is a leapfrog transform of another fullerene. Several basic transformations on maps are defined by means of which the leapfrog and other transformations can be accomplished. 1.
Chemical Graph Theory of Fibonacenes
"... Fibonacenes (zigzag unbranched catacondensed benzenoid hydrocarbons) are a class of polycyclic conjugated systems whose molecular graphs possess remarkable properties, often related with the Fibonacci numbers. This article is a review of the chemical graph theory of fibonacenes, with emphasis on th ..."
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Fibonacenes (zigzag unbranched catacondensed benzenoid hydrocarbons) are a class of polycyclic conjugated systems whose molecular graphs possess remarkable properties, often related with the Fibonacci numbers. This article is a review of the chemical graph theory of fibonacenes, with emphasis on their Kekulestructurerelated and Clarstructurerelated properties.
Polyhedral Combinatorics of Benzenoid Problems
 Lect. Notes Comput. Sci
, 1998
"... Many chemical properties of benzenoid hydrocarbons can be understood in terms of the maximum number of mutually resonant hexagons, or Clar number, of the molecules. Hansen and Zheng (1994) formulated this problem as an integer program and conjectured, based on computational results, that solving the ..."
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Many chemical properties of benzenoid hydrocarbons can be understood in terms of the maximum number of mutually resonant hexagons, or Clar number, of the molecules. Hansen and Zheng (1994) formulated this problem as an integer program and conjectured, based on computational results, that solving the linear programming relaxation always yields integral solutions. We prove this conjecture showing that the constraint matrices of these problems are unimodular. This establishes the integrality of the relaxation polyhedra since the linear programs are in standard form. However, the matrices are not, in general, totally unimodular as is often the case with other combinatorial optimization problems that give rise to integral polyhedra. Similar results are proved for the Fries number, another optimization problem for benzenoids. In a previous paper, Hansen and Zheng (1992) showed that a certain minimum weight cut cover problem defined for benzenoids yields an upper bound for the Clar number and...
Situ Chemical Oxidation of Creosote/Coal Tar Residuals: Experimental and Numerical Investigation
, 2004
"... I herby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final version, as accepted by my examiners. I understand that my thesis may be made electronically available to the public ..."
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I herby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final version, as accepted by my examiners. I understand that my thesis may be made electronically available to the public
Extremal fullerene graphs with the maximum Clar number
, 801
"... A fullerene graph is a cubic 3connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let Fn be a fullerene graph with n vertices. A set H of mutually disjoint hexagons of Fn is a sextet pattern if Fn has a perfect matching which alternates on and off each hexagon in H. The ma ..."
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A fullerene graph is a cubic 3connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let Fn be a fullerene graph with n vertices. A set H of mutually disjoint hexagons of Fn is a sextet pattern if Fn has a perfect matching which alternates on and off each hexagon in H. The maximum cardinality of sextet patterns of Fn is the Clar number of Fn. It was shown that the Clar number is no more than ⌊n−12 6 ⌋. Many fullerenes with experimental evidence attain the upper bound, for instance, C60 and C70. In this paper, we characterize extremal fullerene graphs whose Clar numbers equal n−12 6. By the characterization, we show that there are precisely 18 fullerene graphs with 60 vertices, including C60, achieving the maximum Clar number 8 and we construct all these extremal fullerene graphs.
Cluj and Related Polynomials Applied in Correlating Studies †
, 2006
"... A counting polynomial P(G,x) is a description of a graph property P(G) in terms of a sequence of numbers so that the exponents express the extent of its partitions while the coefficients are related to the frequency of the occurrence of partitions. Basic definitions and properties of Cluj counting p ..."
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A counting polynomial P(G,x) is a description of a graph property P(G) in terms of a sequence of numbers so that the exponents express the extent of its partitions while the coefficients are related to the frequency of the occurrence of partitions. Basic definitions and properties of Cluj counting polynomials CJ(G,x) and their relation with Ω(G,x) and NΩ(G,x) polynomials are presented. Analytical relations for the calculation of such polynomials and their singlenumber descriptors in some classes of planar polyhexes are derived. The ability of these descriptors to predict the boiling point, chromatographic retention index, and resonance energy for some planar polyhex compounds, as well as the toxicity of a set of dibenzofurans, is demonstrated. 1.
On kresonant fullerene graphs ∗
, 801
"... A fullerene graph F is a 3connected 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 Malternating 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 ..."
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A fullerene graph F is a 3connected 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 Malternating 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 Malternating. A fullerene graph F is kresonant 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 2resonant. Further, we show that a 3resonant fullerene graph has at most 60 vertices and construct all nine 3resonant fullerene graphs, which are also kresonant for every integer k> 3. Finally, sextet polynomials of the 3resonant fullerene graphs are computed.
Meanfield resonancetheoretic view of benzenoid networks
"... A simple classically based “mean field ” resonancetheoretic approach is described to anticipate the effects of various modes of electron pairing (as in πbond formation) for general “alternant” benzenoid conjugated πnetworks. This simplified approach avoids generation and manipulation of individua ..."
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A simple classically based “mean field ” resonancetheoretic approach is described to anticipate the effects of various modes of electron pairing (as in πbond formation) for general “alternant” benzenoid conjugated πnetworks. This simplified approach avoids generation and manipulation of individual resonance structures, whence application is facilitated, even for very large systems – so large that there might be hundreds or millions or moles or even substantially greater numbers of resonance structures. Some of the predictions so facilitated for several general circumstances are conveniently manifested as explicit algebraic formulas for numbers of unpaired electrons. The simplicity (and apparent reliability) of these algebraic formulas is emphasized; they involve no more than counts of different “kinds ” of πcenter carbons, e.g., the numbers of πcenters of different functionalities (primary, secondary, or tertiary) which are “starred ” (in an “alternant ” system) as well as those which are “unstarred”. The argumentation also provides some information about rough locations for the (unpaired) spin densities, and ultimately also about the presence of lowlying excited states, magnetic moments, reactivities, and more. In particular these easily used ideas, which are extensions of those already familiar in “classical ” organic chemistry should then aid in nanotechnological developments.
On the sextet polynomial of fullerenes
, 2009
"... We show that the sextet pattern count of every fullerene is strictly smaller than the Kekulé structure count. This proves a conjecture of Zhang and He [J. Math. Chem. 38(3):2005, p. 315–324]. 1 ..."
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We show that the sextet pattern count of every fullerene is strictly smaller than the Kekulé structure count. This proves a conjecture of Zhang and He [J. Math. Chem. 38(3):2005, p. 315–324]. 1
The Clar formulas of a . . .
"... It is shown that the number of Clar formulas of a Kekuléan benzenoid system B is equal to the number of subgraphs of the resonance graph of B isomorphic to the Cl(B)dimensional hypercube, where Cl(B) is the Clar number of B. ..."
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It is shown that the number of Clar formulas of a Kekuléan benzenoid system B is equal to the number of subgraphs of the resonance graph of B isomorphic to the Cl(B)dimensional hypercube, where Cl(B) is the Clar number of B.