2-Domination and Annihilation Numbers

Using information provided by Ryan Pepper and Ermelinda DeLaVina in their paper On the 2-Domination number and Annihilation Number, I developed a new bound on the 2- domination number of trees. An original bound, γ2(G) ≤ (n+n1)/ 2 , had been shown by many other authors. Our goal was to generate a tighter bound in some cases and work towards generating a more general bound on the 2-domination number for all graphs. Throughout the span of this project I generated and proved the bound γ2(T ) ≤ 1/3 (n + 2n1 + n2). To prove this bound I first proved that the 2-domination number of a tree was less than or equal to the sum of two sub-trees formed by the deletion of an edge: γ2(T ) ≤ γ2(T1) + γ2(T2). From there, I proved our bound by showing that a minimum counter-example did not exist. A large portion of the results involves cases where a graph T is considered the minimum counter-example for the sake of contradiction. From there, I showed that if T was a counter-example, then a sub-tree T1 was also a counter-example, meaning that T would no longer be the minimum counter-example. The last portion of the results is a section comparing the two bounds on the 2-domination number with respect to the number of edges and the degrees of those edges

FROM IRREDUNDANCE TO ANNIHILATION: A BRIEF OVERVIEW OF SOME DOMINATION PARAMETERS OF GRAPHS

Durante los últimos treinta años, el concepto de dominación en grafos ha levantado un interés impresionante. Una bibliografía reciente sobre el tópico contiene más de 1200 referencias y el número de definiciones nuevas está creciendo continuamente. En vez de intentar dar un catálogo de todas ellas, examinamos las nociones más clásicas e importantes (tales como dominación independiente, dominación irredundante, k-cubrimientos, conjuntos k-dominantes, conjuntos Vecindad Perfecta, ...) y algunos de los resultados más significativos.   PALABRAS CLAVES: Teoría de grafos, Dominación.   ABSTRACT During the last thirty years, the concept of domination in graphs has generated an impressive interest. A recent bibliography on the subject contains more than 1200 references and the number of new definitions is continually increasing. Rather than trying to give a catalogue of all of them, we survey the most classical and important notions (as independent domination, irredundant domination, k-coverings, k-dominating sets, Perfect Neighborhood sets, ...) and some of the most significant results.   KEY WORDS: Graph theory, Domination

Fuzzy and Neutrosophic Analysis of Periyar's views on Untouchability

This book is organized into four chapters. In Chapter One we just introduce the basic Fuzzy and Neutrosophic tools used in the analysis of the social evil of Untouchability. Since the notion of caste is based on themind, it is appropriate to use Fuzzy and Neutrosophic theory. In Chapter Two we use the opinion of several experts to analyze the various aspects of untouchability. Here, we use the tools that have been described in Chapter One. Fuzzy Directed Graphs and Neutrosophic Directed Graphs of these Fuzzy and Neutrosophic models happen to be very dense. We have represented 16 such graphs in this book. In Chapter Three we give a brief introduction about the life and struggle of Periyar. The fourth and concluding chapter gives observations drawn from our mathematical results based on the Fuzzy and Neutrosophic analysis.Comment: 385 pages, 16 figure

Economic Regulation as Materialization of Development Ideology in Latin America

This article is part of an extensive research project that aims to verify the particularities of social and economic development in Latin America in the late 20th and early 21st centuries We work with the hypothesis that although the countries of the Latin American continent present different degrees of development their economies still remain linked to a type of external regulation that induces them to believe in the possibilities of development without however reversing the levels of dependence on the countries of central capitalism It is therefore an ideology of development that enables topical occasional and specific advances but does not overcome the economic and political subordination of the Latin America In this article we present the two main assumptions of the research firstly we explain what we mean by development ideology and secondly we summarize some aspects that demonstrate how development ideology is expressed in Latin America in the late 20th and early 21st centuries through diversified economic regulation processe

The roles of random boundary conditions in spin systems

Random boundary conditions are one of the simplest realizations of quenched disorder. They have been used as an illustration of various conceptual issues in the theory of disordered spin systems. Here we review some of these result