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

Characterization of reducible hexagons and fast decomposition of elementary benzenoid graphs

A benzenoid graph is a finite connected plane graph with no cut vertices in which every interior region is bounded by a regular hexagon of a side length one. A benzenoid graph G is elementary if every edge belongs to a 1-factor of G. A hexagon h of an elementary benzenoid graph is reducible, if the removal of boundary edges and vertices of h results in an elementary benzenoid graph. We characterize the reducible hexagons of an elementary benzenoid graph. The characterization is the basis for an algorithm which finds the sequence of reducible hexagons that decompose a graph of this class in ▫$O(n^2)$▫ time. Moreover, we present an algorithm which decomposes an elementary benzenoid graph with at most one pericondensed component in linear time

Towards a circular economy: fabrication and characterization of biodegradable plates from sugarcane waste

Bagasse pulp is a promising material to produce biodegradable plates. Bagasse is the fibrous residue that remains after sugarcane stalks are crushed to extract their juice. It is a renewable resource and is widely available in many countries, making it an attractive alternative to traditional plastic plates. Recent research has shown that biodegradable plates made from Bagasse pulp have several advantages over traditional plastic plates. For example, they are more environmentally friendly because they are made from renewable resources and can be composted after use. Additionally, they are safer for human health because they do not contain harmful chemicals that can leach into food. The production process for Bagasse pulp plates is also relatively simple and cost-effective. Bagasse is first collected and then processed to remove impurities and extract the pulp. The pulp is then molded into the desired shape and dried to form a sturdy plate. Overall, biodegradable plates made from Bagasse pulp are a promising alternative to traditional plastic plates. They are environmentally friendly, safe for human health, and cost-effective to produce. As such, they have the potential to play an important role in reducing plastic waste and promoting sustainable practices. Over the years, the world was not paying strict attention to the impact of rapid growth in plastic use. As a result, uncontrollable volumes of plastic garbage have been released into the environment. Half of all plastic garbage generated worldwide is made up of packaging materials. The purpose of this article is to offer an alternative by creating bioplastic goods that can be produced in various shapes and sizes across various sectors, including food packaging, single-use tableware, and crafts. Products made from bagasse help address the issue of plastic pollution. To find the optimum option for creating bagasse-based biodegradable dinnerware in Egypt and throughout the world, researchers tested various scenarios. The findings show that bagasse pulp may replace plastics in biodegradable packaging. As a result of this value-added utilization of natural fibers, less waste and less of it ends up in landfills. The practical significance of this study is to help advance low-carbon economic solutions and to produce secure bioplastic materials that can replace Styrofoam in tableware and food packaging production