Aromatička svojstva potpunih benzenoidnih ugljikovodika

We consider a family of structurally closely related fully-benzenoid hydrocarbons of increasing number of fused benzene rings. Local and global aromatic properties of such molecules are investigated with a particular interest in investigating the role of the finite size of such molecules in modelling the high-polymer or even graphite. An interesting alternation of local properties for benzene rings in a similar environment was observed.Razmatrana je skupina strukturno vrlo sličnih potpunih benzenoidnih ugljikovodika. Studirana su lokalna i globalna svojstva ovih molekula, a naročita je pažnja posvećena ulozi konačne veličine studiranih molekula u modeliranju benzenoidnih polimera ili čak grafita. Opažena je interesantna pojava da lokalna svojstva benzenskih prstenova alterniraju u sličnom okolišu, iako razlike u veličinama opadaju rastom veličine molekule

Topological Properties of Benzenoid Systems. X. Note on a Graph-Theoretical Polynomial of Knop and Trinajstic

A new graph-theoretical polynomial T (G; x) was recently introduced by Knop and Trinajstic\u27:\u27. T (G ; x) differs from the sextet polynomial. The basic mathematical properties of T (G; x) are determined

Partly Olefinic Reference Structure Defined to Evaluate Bond Resonance Energy and the Ring Current It Would Sustain

The concept of bond resonance energy (BRE) is useful for estimating the degree of kinetic stability for a variety of polycyclic π-systems. In evaluating the BRE for a given π bond in a polycyclic π-system, the corresponding partly olefinic reference structure has been defined, in which cyclic conjugation via the π bond is forbidden. The ring-current pattern expected for such a hypothetical π-system were estimated graph-theoretically. Such a ring-current pattern proved very useful to visualizing and characterizing the mode of cyclic conjugation in a partly olefinic π-system

Convexity deficit of benzenoids

In 2012, a family of benzenoids was introduced by Cruz, Gutman, and Rada, which they called convex benzenoids. In this paper we introduce the convexity deficit, a new topological index intended for benzenoids and, more generally, fusenes. This index measures by how much a given fusene departs from convexity. It is defined in terms of the boundary-edges code. In particular, convex benzenoids are exactly the benzenoids having convexity deficit equal to 0. Quasi-convex benzenoids form the family of non-convex benzenoids that are closest to convex, i.e., they have convexity deficit equal to 1. Finally, we investigate convexity deficit of several important families of benzenoids

The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website

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