5 research outputs found
Extremal fullerene graphs with the maximum Clar number
A fullerene graph is a cubic 3-connected plane graph with (exactly 12)
pentagonal faces and hexagonal faces. Let be a fullerene graph with
vertices. A set of mutually disjoint hexagons of is a sextet
pattern if has a perfect matching which alternates on and off each
hexagon in . The maximum cardinality of sextet patterns of is
the Clar number of . It was shown that the Clar number is no more than
. Many fullerenes with experimental evidence
attain the upper bound, for instance, and . In
this paper, we characterize extremal fullerene graphs whose Clar numbers equal
. By the characterization, we show that there are precisely 18
fullerene graphs with 60 vertices, including , 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
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
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
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..