5 research outputs found

    Extremal fullerene graphs with the maximum Clar number

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    A fullerene graph is a cubic 3-connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let FnF_n be a fullerene graph with nn vertices. A set H\mathcal H of mutually disjoint hexagons of FnF_n is a sextet pattern if FnF_n has a perfect matching which alternates on and off each hexagon in H\mathcal H. The maximum cardinality of sextet patterns of FnF_n is the Clar number of FnF_n. It was shown that the Clar number is no more than ⌊n−126⌋\lfloor\frac {n-12} 6\rfloor. Many fullerenes with experimental evidence attain the upper bound, for instance, C60\text{C}_{60} and C70\text{C}_{70}. In this paper, we characterize extremal fullerene graphs whose Clar numbers equal n−126\frac{n-12} 6. By the characterization, we show that there are precisely 18 fullerene graphs with 60 vertices, including C60\text{C}_{60}, achieving the maximum Clar number 8 and we construct all these extremal fullerene graphs.Comment: 35 pages, 43 figure

    Maximum cardinality resonant sets and maximal alternating sets of hexagonal systems

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    AbstractIt is shown that the Clar number can be arbitrarily larger than the cardinality of a maximal alternating set. In particular, a maximal alternating set of a hexagonal system need not contain a maximum cardinality resonant set, thus disproving a previously stated conjecture. It is known that maximum cardinality resonant sets and maximal alternating sets are canonical, but the proofs of these two theorems are analogous and lengthy. A new conjecture is proposed and it is shown that the validity of the conjecture allows short proofs of the aforementioned two results. The conjecture holds for catacondensed hexagonal systems and for all normal hexagonal systems up to ten hexagons. Also, it is shown that the Fries number can be arbitrarily larger than the Clar number

    Two essays in computational optimization: computing the clar number in fullerene graphs and distributing the errors in iterative interior point methods

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    Fullerene are cage-like hollow carbon molecules graph of pseudospherical sym- metry consisting of only pentagons and hexagons faces. It has been the object of interest for chemists and mathematicians due to its widespread application in various fields, namely including electronic and optic engineering, medical sci- ence and biotechnology. A Fullerene molecular, Γ n of n atoms has a multiplicity of isomers which increases as N iso ∼ O(n 9 ). For instance, Γ 180 has 79,538,751 isomers. The Fries and Clar numbers are stability predictors of a Fullerene molecule. These number can be computed by solving a (possibly N P -hard) combinatorial optimization problem. We propose several ILP formulation of such a problem each yielding a solution algorithm that provides the exact value of the Fries and Clar numbers. We compare the performances of the algorithm derived from the proposed ILP formulations. One of this algorithm is used to find the Clar isomers, i.e., those for which the Clar number is maximum among all isomers having a given size. We repeated this computational experiment for all sizes up to 204 atoms. In the course of the study a total of 2 649 413 774 isomers were analyzed.The second essay concerns developing an iterative primal dual infeasible path following (PDIPF) interior point (IP) algorithm for separable convex quadratic minimum cost flow network problem. In each iteration of PDIPF algorithm, the main computational effort is solving the underlying Newton search direction system. We concentrated on finding the solution of the corresponding linear system iteratively and inexactly. We assumed that all the involved inequalities can be solved inexactly and to this purpose, we focused on different approaches for distributing the error generated by iterative linear solvers such that the convergences of the PDIPF algorithm are guaranteed. As a result, we achieved theoretical bases that open the path to further interesting practical investiga- tion

    Polyhedral Combinatorics of Benzenoid Problems

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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..
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