Orientadores: Carlile Campos Lavor, Leonardo Silva de Lima / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T08:31:47Z (GMT). No. of bitstreams: 1 Amaro_BrunoDias_D.pdf: 1369520 bytes, checksum: a36663d5fd23193d66bb22c83cb932aa (MD5) Previous issue date: 2014 / Resumo: A Teoria Espectral de Grafos é um ramo da Matemática Discreta que se preocupa com a relação entre as propriedades algébricas do espectro de certas matrizes associadas a grafos, como a matriz de adjacência, laplaciana ou laplaciana sem sinal e a topologia dos mesmos. Os autovalores e autovetores das matrizes associadas a um grafo são os invariantes que formam o autoespaço de grafos. Em Teoria Espectral de Grafos a conjectura proposta por Brouwer e Haemers, que associa a soma dos k maiores autovalores da matriz Laplaciana de um grafo G com seu número de arestas mais um fator combinatório (que depende do valor k adotado) é uma das questões interessantes e que está em aberto na literatura. Essa mostra diversos trabalhos que tentam provar tal conjectura. Em 2013, Ashraf et al. estenderam essa conjectura para a matriz laplaciana sem sinal e provaram que ela é válida para a soma dos 2 maiores autovalores e que também é válida para todo k, caso o grafo seja regular. Nosso trabalho aborda a versão dessa conjectura para a matriz laplaciana sem sinal. Conseguimos obter uma família de grafos que satisfaz a conjectura para a soma dos 3 maiores autovalores da matriz laplaciana sem sinal e a família de grafos split completo mais uma aresta satisfaz a conjectura para todos os autovalores. Ainda, baseado na desigualdade de Schur, conseguimos mostrar que a soma dos k menores autovalores das matrizes laplaciana e laplaciana sem sinal são limitadas superiormente pela soma dos k menores graus de G / Abstract: The Spectral Graph Theory is a branch of Discrete Mathematics that is concerned with relations between the algebraic properties of spectrum of some matrices associated to graphs, as the Adjacency, Laplacian and signless Laplacian matrices and their respective topologies. The eigenvalues and eigenvectors of matrices associated to graphs are the invariants which constitute the eigenspace of graphs. On Spectral Graph Theory the conjecture proposed by Brouwer and Haemers, associating the sum of k largest eigenvalues of Laplacian matrix of a graph G with its edges numbers plus a combinatorial factor (which depends on the choosed k) is an open interesting question in the Literature. There are several works that attempt to prove this conjecture. In 2013, Ashraf et al. stretch the conjecture out to signless Laplacian matrix and proved that it is true for the sum of the 2 largest eigenvalues of signless Laplacian matrix and it is also true for all k if G is a regular graph. Our work approaches on the version of the conjecture concerning to signless Laplacian matrix. We could obtain a family of graphs which satisfies the conjecture for the sum of the 3 largest eigenvalues of signless Laplacian matrix and we prove that the family of complete split graphs plus one edge satisfies the Conjecture for all eigenvalues. Moreover, based on Schur's inequality, we could show that the sum of the k smallest eigenvalues of Laplacian and signless Laplacian matrices are bounded by the sum of the k smallest degrees of G / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

Orientadores: Denis José Schiozer, Arnaud Lange / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica e Instituto de Geociências / Made available in DSpace on 2018-08-26T16:46:39Z (GMT). No. of bitstreams: 1 Lima_Alexandrede_M.pdf: 3755455 bytes, checksum: 9ceb17ac0ecaaecbc45e059c8549115a (MD5) Previous issue date: 2014 / Resumo: A maior parte das reservas provadas de óleo convencional no mundo está contida nos reservatórios carbonáticos, as quais, em sua maioria, apresentam fraturas responsáveis por impactarem no fluxo do reservatório. Estas descontinuidades conhecidas como fraturas são encontradas na natureza em diversas escalas e, dependendo do tamanho, podem apresentar dificuldades para serem caracterizadas e modeladas matematicamente. Para exemplificar, pode ser citada a complexidade intrínseca à caracterização de fraturas subsísmicas para modelar objetos em escala menor do que a escala de dados de sísmica e poço. De maneira geral, as fraturas sempre foram um desafio devido a diversos motivos tais como o acréscimo no tempo computacional nas simulações e as dificuldades na caracterização. Estas preocupações se agravam pelo fato de que, na maior parte das vezes, os estudos em engenharia já são complexos, iterativos e consomem elevado tempo até sua finalização como, por exemplo, no processo de ajuste de histórico. Com a intenção de auxiliar e reduzir o tempo despendido nestes estudos é proposta a construção de uma ferramenta rápida capaz de selecionar modelos através do uso da teoria dos Grafos, antes de partir diretamente para onerosas simulações de reservatórios fraturados. Assim, desenvolve-se uma metodologia de análise de conectividade entre poços e também entre poço-reservatório baseada na representação de modelos de reservatórios através da teoria dos Grafos e do uso de seus algoritmos. Esta metodologia é utilizada em quatro aplicações distintas: (1) seleção inicial de modelos estáticos, (2) validação da relação entre conectividade e tempo de irrupção de água, (3) auxílio no processo de ajuste de histórico e (4) melhoria da eficiência de varrido / Abstract: Most proven conventional oil reserves in the world are contained in carbonate reservoirs, which mostly of them have fractures that impact on reservoir dynamic behavior. These discontinuities well-known as fractures are found in nature on several scales and, depending on theirs size, they can present many difficulties to be characterized and modeled mathematically. As an example that can be mentioned is the intrinsic complexity of subseismic fractures characterization to model objects on a smaller scale than the scale of seismic and well data. Overall, fractures have always been a challenge because of various reasons like the increasing in computational time simulating or the difficulties faced to characterize them. These concerns are related to the fact that most of the engineering studies are already complex, iterative and consume large time until its conclusion as, for example, the history matching process. In order to assist and reduce the time spent on these studies is proposed to build a fast tool able to select models through the use of the theory of Graphs, before leaving directly to costly fractured reservoir simulations. Thus, it is developed a methodology of connectivity analysis between wells and also between well and all reservoir, which is based on the representation of reservoir models by Graph theory and the use of its algorithms. This methodology is employed in four different applications: (1) initial selection of static models, (2) validation of the relationship between connectivity and breakthrough time, (3) to assist the history matching process and (4) for improving the sweep efficiency / Mestrado / Reservatórios e Gestão / Mestre em Ciências e Engenharia de Petróleo

Orientador: Luiz Camolesi Júnior / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Tecnologia / Made available in DSpace on 2018-08-27T04:46:35Z (GMT). No. of bitstreams: 1 Fonseca_AdrianedeMartini_M.pdf: 1663096 bytes, checksum: 54c4089adea68e67cb7ab43b4b46dfdf (MD5) Previous issue date: 2015 / Resumo: A informação produzida atualmente apresenta crescimento em volume e complexidade, representando um desafio tecnológico que demanda mais do que a atual estrutura de Bancos de Dados Relacionais pode oferecer. Tal fato estimula o uso de diferentes formas de armazenamento, como Bancos de Dados baseados em Grafos (BDG). Os atuais Bancos de Dados baseados em Grafos são adaptados para suportar automaticamente a evolução do banco de dados, mas não fornecem recursos adequados para a organização da informação. Esta função é deixada a cargo das aplicações que acessam o banco de dados, comprometendo a integridade dos dados e sua confiabilidade. O objetivo deste trabalho é a definição de regras de refatoração para auxiliar o gerenciamento da evolução de Bancos de Dados baseados em Grafos. As regras apresentadas neste trabalho são adaptações e extensões de regras de refatoração consolidadas para bancos de dados relacionais para atender às características dos Bancos de Dados baseado em Grafos. O resultado deste trabalho é um catálogo de regras que poderá ser utilizado por desenvolvedores de ferramentas de administração de bancos de dados baseados em grafos para garantir a integridade das operações de evolução de esquemas de dados e consequentemente dos dados relacionados / Abstract: The information produced nowadays does not stop growing in volume and complexity, representing a technological challenge which demands more than the relational model for databases can currently offer. This situation stimulates the use of different forms of storage, such as Graph Databases. Current Graph Databases allow automatic database evolution, but do not provide adequate resources for the information organization. This is mostly left under the responsibility of the applications which access the database, compromising the data integrity and reliability. The goal of this work is the definition of refactoring rules to support the management of the evolution of Graph Databases. The rules presented in this document are adaptations and extensions of the existent refactoring rules for relational databases to meet the requirements of the Graph Databases features. The result of this work is a catalog of refactoring rules that can be used by developers of graph database management tools to guarantee the integrity of the operations of database evolution / Mestrado / Tecnologia e Inovação / Mestra em Tecnologia

Considering the Chinese Postman Problem, in which a mailman must deliver mail to houses in a neighborhood. The mailman must cover each side of the street that has houses, at least once. The focus of this paper is our attempt to discover the optimal path, or the least number of times each street is walked. The integration of algorithms from graph theory and operations research form the method used to explain solutions to the Chinese Postman Problem.

This project deals with the automorphism group G of a Steiner system S (3, 4, 10). S₁₀, the symmetrical group of degree 10, acts transitively on T, the set of all Steiner systems with parameters 3, 4, 10. The purpose of this project is to study the action of S₁₀ on cosets of G. This will be achieved by means of a graph of S₁₀ on T x T. The orbits of S₁₀ on T x T are in one-one correspondence with the orbits of G, the stabilizer of an S [e] T on T.

Vizing conjectured in 1963 that the domination number of the Cartesian product of two graphs is at least the product of their domination numbers; this remains one of the biggest open problems in the study of domination in graphs. Several partial results have been proven, but the conjecture has yet to be proven in general. The purpose of this thesis was to study Vizing's conjecture, related results, and open problems related to the conjecture. We give a survey of classes of graphs that are known to satisfy the conjecture, and of Vizing-like inequalities and conjectures for different types of domination and graph products. We also give an improvement of the Clark-Suen inequality. Some partial results about fair domination are presented, and we summarize some open problems related to Vizing's conjecture.

Dans cette thèse, nous étudions des problèmes différents de coloration maximale dans les graphes sommet-colorés. Nous nous concentrons sur la recherche des structures avec le nombre maximal possible de couleurs par des algorithmes en temps polynomial, nous donnons aussi la preuve des problèmes NP-difficiles pour des graphes spécifiques. En particulier, nous étudions d’abord le problème de l’appariement coloré maximum. Nous montrons que ce problème peut être résolu efficacement en temps polynomial. En plus, nous considérons également une version spécifique de ce problème, à savoir l’appariement tropical, qui consiste à trouver un appariement contenant toutes les couleurs du graphe original. De même, un algorithme de temps polynomial est également fourni pour le problème de l’appariement tropical avec la cardinalité minimale et le problème de l’appariement tropical maximum avec la cardinalité minimale. Ensuite, nous étudions le problème des chemins colorés maximum. Il existe deux versions pour ce problème: le problème de plus court chemin tropical, c’est-à-dire de trouver un chemin tropical avec le poids total minimum et le problème de plus longue chemin coloré, à savoir, trouver un chemin avec un nombre maximum possible de couleurs. Nous montrons que les deux versions de ce problème sont NP-difficile pour un graphe orienté acyclique, graphes de cactus et graphes d'intervalles où le problème de plus long chemin est facile. De plus, nous fournissons également un algorithme de paramètre fixe pour le premier dans les graphes généraux et plusieurs algorithmes de temps polynomiaux pour le second dans les graphes spécifiques, y compris les graphes des chaîne bipartites, graphes de seuil, arborescences, graphes des blocs et graphes d'intervalles appropriés. Ensuite, nous considérons le problème des cycles colorés maximum. Nous montrons d'abord que le problème est NP-difficile même pour des graphes simples tels que des graphes divisés, des graphes bi-connecteurs et des graphes d'intervalles. Nous fournissons ensuite des algorithmes de temps polynomial pour les classes de graphes de seuil et graphes des chaîne bipartites et graphes d'intervalles appropriés. Plus tard, nous étudions le problème des cliques colorées maximum. Nous montrons tout d’abord que le problème est NP-difficile même pour plusieurs cas où le problème de clique maximum est facile, comme des graphes complémentaires des graphes de permutation bipartite, des graphes complémentaires de graphes convexes bipartites et des graphes de disques unitaires, et aussi pour des graphes sommet-colorées appropriés. Ensuite, nous proposons un algorithme paramétré XP et des algorithmes de temps polynomial pour les classes de graphes complémentaires de graphes en chaîne bipartites, des graphes multipartites complets et des graphes complémentaires de graphes cycles. Enfin, nous nous concentrons sur le problème des stables (ensembles indépendants) colorés maximum. Nous montrons d’abord que le problème est NP-difficile même dans certains cas où le problème de stable maximum est facile, tels que les co-graphes et les graphes des P₅-gratuit. Ensuite, nous fournissons des algorithmes de temps polynomial pour les graphes de grappes, et les arbres. / In this thesis, we study different maximum colorful problems in vertex-colored graphs. We focus on finding structures with the possible maximum number of colors by efficient polynomial-time algorithms, or prove these problems as NP-hard for specific graphs. In particular, we first study the maximum colorful matching problem. We show that this problem can be efficiently solved in polynomial time. Moreover, we also consider a specific version of this problem, namely tropical matching, that is to find a matching containing all colors of the original graph, if any. Similarly, a polynomial time algorithm is also provided for the problem of tropical matching with the minimum cardinality and the problem of maximal tropical matching with the minimum cardinality. Then, we study the maximum colorful paths problem. There are two versions for this problem: the shortest tropical path problem, i.e., finding a tropical path with the minimum total weight, and the maximum colorful path problem, i.e., finding a path with the maximum number of colors possible. We show that both versions of this problem are NP-hard for directed acyclic graphs, cactus graphs and interval graphs where the longest path problem is easy. Moreover, we also provide a fixed parameter algorithm for the former in general graphs and several polynomial time algorithms for the latter in specific graphs, including bipartite chain graphs, threshold graphs, trees, block graphs, and proper interval graphs. Next we consider the maximum colorful cycles problem. We first show that the problem is NP-hard even for simple graphs such as split graphs, biconnected graphs, interval graphs. Then we provide polynomial-time algorithms for classes of threshold graphs and bipartite chain graphs and proper interval graphs. Later, we study the maximum colorful cliques problem. We first show that the problem is NP-hard even for several cases where the maximum clique problem is easy, such as complement graphs of bipartite permutation graphs, complement graphs of bipartite convex graphs, and unit disk graphs, and also for properly vertex-colored graphs. Next, we propose a XP parameterized algorithm and polynomial-time algorithms for classes of complement graphs of bipartite chain graphs, complete multipartite graphs and complement graphs of cycle graphs. Finally, we focus on the maximum colorful independent set problem. We first prove that the problem is NP-hard even for some cases where the maximum independent set problem is easy, such as cographs and P₅-free graphs. Next, we provide polynomial time algorithms for cluster graphs and trees.

This thesis describes the development of the SmartGraph, an AI enabled graph database. The need for such a system has been independently recognized in the isolated fields of graph databases, graph computing, and computational graph deep learning systems, such as TensorFlow. Though prior works have investigated some relationships between these fields, we believe that the SmartGraph is the first system designed from conception to incorporate the most significant and useful characteristics of each. Examples include the ability to store graph structured data, run analytics natively on this data, and run gradient descent algorithms. It is the synergistic aspects of combining these fields that provide the most novel results presented in this dissertation. Key among them is how the notion of “graph querying” as used in graph databases can be used to solve a problem that has plagued deep learning systems since their inception; rather than attempting to embed graph structured datasets into restrictive vector spaces, we instead allow the deep learning functionality of the system to natively perform graph querying in memory during optimization as a way of interpreting (and learning) the graph. This results in a concept of natural and interpretable processing of graph structured data. Graph computing systems have traditionally used distributed computing across multiple compute nodes (e.g. separate machines connected via Ethernet or internet) to deal with large-scale datasets whilst working sequentially on problems over entire datasets. In this dissertation, we outline a distributed graph computing methodology that facilitates all the above capabilities (even in an environment consisting of a single physical machine) while allowing for a workflow more typical of a graph database than a graph computing system; massive concurrent access allowing for arbitrarily asynchronous execution of queries and analytics across the entire system. Further, we demonstrate how this methodology is key to the artificial intelligence capabilities of the system.

To determine the function of a protein, we must know its 3-dimensional structure, which can be difficult to ascertain. Currently, predictive models are used to determine the structure of a protein from its sequence, but these models do not always predict the correct structure. To this end we use a nested graph model along with weighted invariants to minimize the errors and improve the accuracy of a predictive model to determine if we have the correct structure for a protein.

Freshness of information has become of high importance with the emergence of many real- time applications like monitoring systems and communication networks. The main idea behind all of these scenarios is the same, there exists at least a monitor of some process to which the monitor does not have direct access. Rather, the monitor indirectly receives updates over time from a source that can observe the process directly. The common main goal in these scenarios is to guarantee that the updates at the monitor side are as fresh as possible. However, due to the contention among the nodes in the network over limited channel resources, it takes some random time for the updates before they are received by the monitor. These applications have motivated a line of research studying the Age of Information (AoI) as a new performance metric that captures timeliness of information. The first part of this dissertation focuses on the AoI problem in general multi-source multi-hop status update networks with slotted transmissions. Fundamental lower bounds on the instantaneous peak and average AoI are derived under general interference constraints. Explicit algorithms are developed that generate scheduling policies for status update dissem- ination throughout the network for the class of minimum-length periodic schedules under global interference constraints. Next, we study AoI in multi-access channels, where a number of sources share the same server with exponentially distributed service times to communicate to a monitor. Two cases depending on the status update arrival rates at the sources are considered: (i) random arrivals based on the Poisson point process, and (ii) active arrivals where each source can generate an update at any point in time. For each case, closed-form expressions are derived for the average AoI as a function of the system parameters. Next, the effect of energy harvesting on the age is considered in a single-source single- monitor status update system that has a server with a finite battery capacity. Depending on the server’s ability to harvest energy while a packet is in service, and allowing or blocking the newly-arriving packets to preempt a packet in service, average AoI expressions are derived. The results show that preemption of the packets in service is sub-optimal when the energy arrival rate is lower than the status update arrival rate. Finally, the age of channel state information (CSI) is studied in fully-connected wire- less networks with time-slotted transmissions and time-varying channels. A framework is developed that accounts for the amount of data and overhead in each packet and the CSI disseminated in the packet. Lower bounds on the peak and average AoI are derived and a greedy protocol that schedules the status updates based on minimizing the instantaneous average AoI is developed. Achievable average AoI is derived for the class of randomized CSI dissemination schedules.