Planar k-cycle resonant graphs with k=1,2

AbstractA connected graph is said to be k-cycle resonant if, for 1⩽t⩽k, any t disjoint cycles in G are mutually resonant, that is, there is a perfect matching M of G such that each of the t cycles is an M-alternating cycle. The concept of k-cycle resonant graphs was introduced by the present authors in 1994. Some necessary and sufficient conditions for a graph to be k-cycle resonant were also given. In this paper, we improve the proof of the necessary and sufficient conditions for a graph to be k-cycle resonant, and further investigate planar k-cycle resonant graphs with k=1,2. Some new necessary and sufficient conditions for a planar graph to be 1-cycle resonant and 2-cycle resonant are established

Recursive formulae for enumeration of LM-conjugated circuits in structurally related benzenoid hydrocarbons

The linearly independent and minimal conjugated (LM-conjugated) circuits of benzenoid hydrocarbons play the central role in the conjugated circuit model. For a general case, the enumeration of LM-conjugated circuits may be tedious as it requires construction of all Kekule structures. In our previous work, a recursive method for enumeration of LM-conjugated circuits of benzenoid hydrocarbons was established. In this paper, we further extend the recursive formulae for enumerations of LM-conjugated circuits for both catacondensed benzenoid hydrocarbons and some families of structurally related pericondensed benzenoid hydrocarbons

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

Analytical expressions for the count of LM-conjugated circuits of benzenoid hydrocarbons

A recursive method for enumeration of linearly independent and minimal conjugated circuits of benzenoid hydrocarbons had previously been given which is valid for several classes of benzenoid hydrocarbons. In the present article, the properties and constructions of unique minimal conjugated circuits and pairs of minimal conjugated circuits of a ring s in a benzenoid hydrocarbon B are investigated. An analytical expression for the count of LM-conjugated circuits of B is given which is based on the counts of Kekule structures of selected subgraphs of B. By using the method, the LMC expression of any benzenoid hydrocarbon can be obtained. (C) 1996 John Wiley & Sons, Inc

Estimation method for the thermochemical properties of polycyclic aromatic molecules

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2005.Includes bibliographical references.Polycyclic aromatic molecules, including polycyclic aromatic hydrocarbons (PAHs) have attracted considerable attention in the past few decades. They are formed during the incomplete combustion of hydrocarbon fuels and are precursors of soot. Some PAHs are known carcinogens, and control of their emissions is an important issue. These molecules are found in many materials, including coal, fuel oils, lubricants, and carbon black. They are also implicated in the formation of fullerenes, one of the most. chemically versatile class of molecules known. Clearly, models that provide predictive capability for their formation and growth are highly desirable. Thlermochemical properties of the species in the model are often the most important parameter, particularly for high temperature processes such as the formation of PAH and other aromatic molecules. Thermodynamic consistency requires that reverse rate constants be calculated from the forward rate constants and from the equilibrium constants. The later are obtained from the thermochemical properties of reactants and products. The predictive ability of current kinetic models is significantly limited by the scarcity of available thermochemical data.(cont.) In this work we present the development of a Bond-Centered Group Additivity method for the estimation of the thermochemical properties of polycyclic aromatic molecules, including PAHs, molecules with the furan substructure, molecules with triple bonds, substituted PAHs, and radicals. This method is based on thermochemical values of about two hundred polycyclic aromatic molecules and radicals obtained from quantum chemical calculations at the B3LYP/6-31G(d) level. A consistent set of homodesmic reactions has been developed to accurately calculate the heat of formation from the absolute energy. The entropies calculated from the B3LYP/6-31G(d) vibrational frequencies are shown to be at least as reliable as the few available experimental values. This new Bond-Centered Group Additivity method predicts the thermochemistry of C₆₀ and C₇₀ fullerenes, as well as smaller aromatic molecules, with accuracy comparable to both experiments and the best quantum calculations. This Bond-Centered Group Additivity method is shown to extrapolate reasonably to infinite graphene sheets.(cont.) The Bond-Centered Group Additivity method has been implemented into a computer code within the automatic Reaction Mechanism Generation software (RMG) developed in our group. The database has been organized as a tree structure, making its maintenance and possible extension very straightforward. This computer code allows the fast and easy use of this estimation method by non-expert users. Moreover, since it is incorporated into RMG, it will allow users to generate reaction mechanisms that include aromatic molecules whose thermochemical properties are calculated using the Bond-Centered Group Additivity method. Exploratory equilibrium studies were performed (l. Equilibrium concentrations of individual species depend strongly on the thermochemistry of the individual species, emphasizing the importance of consistent thermochemistry for all the species involved in the calculations. Equilibrium calculations can provide many interesting insights into the relationship between PAH and fullerenes in combustion.by Joanna Yu.Ph.D