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

A network flow approach to a common generalization of Clar and Fries numbers

Clar number and Fries number are two thoroughly investigated parameters of plane graphs emerging from mathematical chemistry to measure stability of organic molecules. We consider first a common generalization of these two concepts for bipartite plane graphs, and then extend it to a framework on general (not necessarily planar) directed graphs. The corresponding optimization problem can be transformed into a maximum weight feasible tension problem which is the linear programming dual of a minimum cost network flow (or circulation) problem. Therefore the approach gives rise to a min-max theorem and to a strongly polynomial algorithm that relies exclusively on standard network flow subroutines. In particular, we give the first network flow based algorithm for an optimal Fries structure and its variants

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