Thesis (Ph. D.)--West Virginia University, 2005. / Title from document title page. Document formatted into pages; contains vi, 49 p. : ill. Includes abstract. Includes bibliographical references (p. 47-49).

Thesis (M.S.)--West Virginia University, 2005. / Title from document title page. Document formatted into pages; contains v, 17 p. : ill. Includes abstract. Includes bibliographical references (p. 17).

Thesis (Ph. D.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains vi, 52 p. : ill. Includes abstract. Includes bibliographical references (p. 50-52).

Thesis (M.Sc.). / Written for the Dept. of Mathematics and Statistics. Title from title page of PDF (viewed 2008/04/12). Includes bibliographical references.

Thesis (Ph.D.) - Carleton University, 2002. / Includes bibliographical references (p. 98-100). Also available in electronic format on the Internet.

In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameter-constrained minimum spanning tree (DCMST) of a given undirected, edge-weighted graph, G, is the smallest-weight spanning tree of all spanning trees of G which contain no path with more than k edges, where k is a given positive integer. The problem of finding a DCMST is NP-complete for all values of k; 4 ≤ k ≤ (n - 2), except when all edge-weights are identical. A DCMST is essential for the efficiency of various distributed mutual exclusion algorithms, where it can minimize the number of messages communicated among processors per critical section. It is also useful in linear lightwave networks, where it can minimize interference in the network by limiting the traffic in the network lines. Another practical application requiring a DCMST arises in data compression, where some algorithms compress a file utilizing a data-structure, and decompress a path in the tree to access a record. A DCMST helps such algorithms to be fast without sacrificing a lot of storage space. We present a survey of the literature on the DCMST problem, study the expected diameter of a random labeled tree, and present five new polynomial-time algorithms for an approximate DCMST. One of our new algorithms constructs approximate DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to the tree in each stage of the construction. Three other new algorithms start with an unconstrained minimum spanning tree, and iteratively refine it into an approximate DCMST. We also present ab algorithm designed for the special case when the diameter is required to be no more than 4. Such a diameter-4 tree is also used for evaluating the quality of other algorithms. All five algorithms were implemented on a PC, and four of them were also parallelized and implemented on a massively parallel machine-the MasPar MP-1. We discuss convergence, relative merits, and implementation of these heuristics. Our extensive empirical study shows that the heuristics produce good solutions for a wide variety of inputs.

In this dissertation, we are interested in studying several parameters of graphs and understanding their extreme values. We begin in Chapter~$2$ with a question on edge colouring. When can a partial proper edge colouring of a graph of maximum degree $\Delta$ be extended to a proper colouring of the entire graph using an `optimal' set of colours? Albertson and Moore conjectured this is always possible provided no two precoloured edges are within distance $2$. The main result of Chapter~$2$ comes close to proving this conjecture. Moreover, in Chapter~$3$, we completely answer the previous question for the class of planar graphs. Next, in Chapter~$4$, we investigate some Ramsey theoretical problems. We determine exactly what minimum degree a graph $G$ must have to guarantee that, for any two-colouring of $E(G)$, we can partition $V(G)$ into two parts where each part induces a connected monochromatic subgraph. This completely resolves a conjecture of Bal and Debiasio. We also prove a `covering' version of this result. Finally, we study another variant of these problems which deals with coverings of a graph by monochromatic components of distinct colours. The following saturation problem proposed by Barrus, Ferrara, Vandenbussche, and Wenger is considered in Chapter~$5$. Given a graph $H$ and a set of colours $\{1,2,\ldots,t\}$ (for some integer $t\geq |E(H)|$), we define $sat_{t}(n, R(H))$ to be the minimum number of $t$-coloured edges in a graph on $n$ vertices which does not contain a rainbow copy of $H$ but the addition of any non-edge in any colour from $\{1,2,\ldots,t\}$ creates such a copy. We prove several results concerning these extremal numbers. In particular, we determine the correct order of $sat_{t}(n, R(H))$, as a function of $n$, for every connected graph $H$ of minimum degree greater than $1$ and for every integer $t\geq e(H)$. In Chapter~$6$, we consider the following question: under what conditions does a Hamiltonian graph on $n$ vertices possess a second cycle of length at least $n-o(n)$? We prove that the `weak' assumption of a minimum degree greater or equal to $3$ guarantees the existence of such a long cycle. We solve two problems related to majority colouring in Chapter~$7$. This topic was recently studied by Kreutzer, Oum, Seymour, van der Zypen and Wood. They raised the problem of determining, for a natural number $k$, the smallest positive integer $m = m(k)$ such that every digraph can be coloured with $m$ colours, where each vertex has the same colour as at most a proportion of $\frac{1}{k}$ of its out-neighbours. Our main theorem states that $m(k) \in \{2k-1, 2k\}$. We study the following problem, raised by Caro and Yuster, in Chapter~$8$. Does every graph $G$ contain a `large' induced subgraph $H$ which has $k$ vertices of degree exactly $\Delta(H)$? We answer in the affirmative an approximate version of this question. Indeed, we prove that, for every $k$, there exists $g(k)$ such that any $n$ vertex graph $G$ with maximum degree $\Delta$ contains an induced subgraph $H$ with at least $n-g(k)\sqrt{\Delta}$ vertices such that $V(H)$ contains at least $k$ vertices of the same degree $d \ge \Delta(H)-g(k)$. This result is sharp up to the order of $g(k)$. %Subsequently, we investigate a concept called $\textit{path-pairability}$. A graph is said to be path-pairable if for any pairing of its vertices there exist a collection of edge-disjoint paths routing the the vertices of each pair. A question we are concerned here asks whether every planar path pairable graph on $n$ vertices must possess a vertex of degree linear in $n$. Indeed, we answer this question in the affirmative. We also sketch a proof resolving an analogous question for graphs embeddable on surfaces of bounded genus. Finally, in Chapter~$9$, we move on to examine $k$-linked tournaments. A tournament $T$ is said to be $k$-linked if for any two disjoint sets of vertices $\{x_1,\ldots ,x_k\}$ and $\{y_1,\dots,y_k\}$ there are directed vertex disjoint paths $P_1,\dots, P_k$ such that $P_i$ joins $x_i$ to $y_i$ for $i = 1,\ldots, k$. We prove that any $4k$ strongly-connected tournament with sufficiently large minimum out-degree is $k$-linked. This result comes close to proving a conjecture of Pokrovskiy.

Uma imagem engloba informação que precisa ser organizada para interpretar e compreender seu conteúdo. Existem diversas técnicas computacionais para extrair a principal informação de uma imagem e podem ser divididas em três áreas: análise de cor, textura e forma. Uma das principais delas é a análise de forma, por descrever características de objetos baseadas em seus pontos fronteira. Propomos um método de caracterização de imagens, por meio da análise de forma, baseada nas propriedades espectrais do laplaciano em grafos. O procedimento construiu grafos G baseados nos pontos fronteira do objeto, cujas conexões entre vértices são determinadas por limiares T_l. A partir dos grafos obtêm-se a matriz de adjacência A e a matriz de graus D, as quais definem a matriz Laplaciana L=D -A. A decomposição espectral da matriz Laplaciana (autovalores) é investigada para descrever características das imagens. Duas abordagens são consideradas: a) Análise do vetor característico baseado em limiares e a histogramas, considera dois parâmetros o intervalo de classes IC_l e o limiar T_l; b) Análise do vetor característico baseado em vários limiares para autovalores fixos; os quais representam o segundo e último autovalor da matriz L. As técnicas foram testada em três coleções de imagens: sintéticas (Genéricas), parasitas intestinais (SADPI) e folhas de plantas (CNShape), cada uma destas com suas próprias características e desafios. Na avaliação dos resultados, empregamos o modelo de classificação support vector machine (SVM), o qual avalia nossas abordagens, determinando o índice de separação das categorias. A primeira abordagem obteve um acerto de 90 % com a coleção de imagens Genéricas, 88 % na coleção SADPI, e 72 % na coleção CNShape. Na segunda abordagem, obtém-se uma taxa de acerto de 97 % com a coleção de imagens Genéricas; 83 % para SADPI e 86 % no CNShape. Os resultados mostram que a classificação de imagens a partir do espectro do Laplaciano, consegue categorizá-las satisfatoriamente. / An image includes information that needs to be organized to interpret and understand its contents. There are several computational techniques to extract the main information of images and are divided into three areas: color, texture and shape analysis. One of the main of them is shape analysis, since it describes objects getting main features based on reference points, usually border points. This dissertation proposes a shape analysis method based on the spectral properties of the Laplacian in graphs to represent images. The procedure builds G graphs based on object border points, whose connections between vertices are determined by thresholds T_l. From graphs G we obtain the adjacency matrix A and matrix degrees D, which define the Laplacian matrix L=D -A. Thus, spectral decomposition of the Laplacian matrix (eigenvalues) is investigated to describe image features. Two approaches are considered: a)Analysis of feature vector based on thresholds and histograms, it considers two parameters, classes range IC_l and threshold T_l; b) Analysis of feature vector based on multiple linear for fixed eigenvalues, which represents the second and final eigenvalue matrix L. The techniques were tested in three image datasets: synthetic (Generic), human intestinal parasites (SADPI) and plant leaves (CNShape), each of these with its own features and challenges. Afterwards to evaluate our results, we used the classification model Support Vector Machine (SVM) to evaluate our approaches, determining the percentage of separation of categories. The first approach achieved 90 % of precision with the Generic image dataset, 88 % in SADPI dataset, and 72 % in CNShape dataset. In the second approach, it obtains 97 % of precision with the Generic image dataset, 83 % for SADPI and 86 % in CNShape respectively. The results show that the classification of images from the Laplacian spectrum can categorize them satisfactorily.

Uma imagem engloba informação que precisa ser organizada para interpretar e compreender seu conteúdo. Existem diversas técnicas computacionais para extrair a principal informação de uma imagem e podem ser divididas em três áreas: análise de cor, textura e forma. Uma das principais delas é a análise de forma, por descrever características de objetos baseadas em seus pontos fronteira. Propomos um método de caracterização de imagens, por meio da análise de forma, baseada nas propriedades espectrais do laplaciano em grafos. O procedimento construiu grafos G baseados nos pontos fronteira do objeto, cujas conexões entre vértices são determinadas por limiares T_l. A partir dos grafos obtêm-se a matriz de adjacência A e a matriz de graus D, as quais definem a matriz Laplaciana L=D -A. A decomposição espectral da matriz Laplaciana (autovalores) é investigada para descrever características das imagens. Duas abordagens são consideradas: a) Análise do vetor característico baseado em limiares e a histogramas, considera dois parâmetros o intervalo de classes IC_l e o limiar T_l; b) Análise do vetor característico baseado em vários limiares para autovalores fixos; os quais representam o segundo e último autovalor da matriz L. As técnicas foram testada em três coleções de imagens: sintéticas (Genéricas), parasitas intestinais (SADPI) e folhas de plantas (CNShape), cada uma destas com suas próprias características e desafios. Na avaliação dos resultados, empregamos o modelo de classificação support vector machine (SVM), o qual avalia nossas abordagens, determinando o índice de separação das categorias. A primeira abordagem obteve um acerto de 90 % com a coleção de imagens Genéricas, 88 % na coleção SADPI, e 72 % na coleção CNShape. Na segunda abordagem, obtém-se uma taxa de acerto de 97 % com a coleção de imagens Genéricas; 83 % para SADPI e 86 % no CNShape. Os resultados mostram que a classificação de imagens a partir do espectro do Laplaciano, consegue categorizá-las satisfatoriamente. / An image includes information that needs to be organized to interpret and understand its contents. There are several computational techniques to extract the main information of images and are divided into three areas: color, texture and shape analysis. One of the main of them is shape analysis, since it describes objects getting main features based on reference points, usually border points. This dissertation proposes a shape analysis method based on the spectral properties of the Laplacian in graphs to represent images. The procedure builds G graphs based on object border points, whose connections between vertices are determined by thresholds T_l. From graphs G we obtain the adjacency matrix A and matrix degrees D, which define the Laplacian matrix L=D -A. Thus, spectral decomposition of the Laplacian matrix (eigenvalues) is investigated to describe image features. Two approaches are considered: a)Analysis of feature vector based on thresholds and histograms, it considers two parameters, classes range IC_l and threshold T_l; b) Analysis of feature vector based on multiple linear for fixed eigenvalues, which represents the second and final eigenvalue matrix L. The techniques were tested in three image datasets: synthetic (Generic), human intestinal parasites (SADPI) and plant leaves (CNShape), each of these with its own features and challenges. Afterwards to evaluate our results, we used the classification model Support Vector Machine (SVM) to evaluate our approaches, determining the percentage of separation of categories. The first approach achieved 90 % of precision with the Generic image dataset, 88 % in SADPI dataset, and 72 % in CNShape dataset. In the second approach, it obtains 97 % of precision with the Generic image dataset, 83 % for SADPI and 86 % in CNShape respectively. The results show that the classification of images from the Laplacian spectrum can categorize them satisfactorily.

D’un côté, la transition écologique et les enjeux de développement durable sont de nos jours une réalité que l’on ne peut ignorer compte tenu des impacts négatifs des activités humaines sur leurs environnements. De l’autre côté, une numérisation toujours plus importante de ces environnements entraîne la génération de volumes massifs de traces numériques, qui sont autant d’indices sur le monde dans lequel vivent les acteurs de ces activités. Une difficulté non négligeable existe pour comprendre les tenants et aboutissants faisant que d’une activité à une autre, l’impact sur l’environnement mesuré dans ces travaux de recherche à travers le concept d’Emprise Énergétique (EmE) n’est pas le même. Notre approche considère l’identification sur la base de ces traces numériques, d’activité d’entités humaines et non humaines. L’instanciation de ces dernières au sein de pratiques mobilise des ressources (physiques et virtuelles) en plus ou moins grand nombre. Leurs modélisations permettraient de mieux appréhender les enjeux liés à la transition écologique. Identifier sur la base d’indicateurs quantifiables les pratiques ayant un impact réduit sur l’environnement serait une piste permettant de contribuer à cette transition. Ces pratiques, au sens de coordination de multiples entités hétérogènes dans le temps et l’espace, peuvent être formalisées sous forme de structures d’activités multidimensionnelles à l’aide de la théorie de l’Agencement et d’un ensemble d’outils mathématiques (Complexes Simpliciaux, Hypernetworks). Ces travaux de recherche tentent de modéliser le phénomène d’activité humaine et non humaine en s’appuyant sur la caractérisation du contexte de celles-ci à partir de données massives. Ces agencements sont calculés et représentés dans une application (IMhoTEP) ayant pour but de construire ces structures complexes non pas sur des catégorisations faites a priori des entités, mais en se focalisant sur les relations que celles-ci entretiennent dans plusieurs dimensions. L’objectif final est de proposer un outil d’accompagnement à la transition écologique à destination des acteurs participant à des activités induisant la consommation, voire la production de ressources. Ces travaux de recherche en informatique s’appuient sur la numérisation continue des espaces et particulièrement les espaces urbains (Smart City, Internet of Everything). / On one hand, the ecological transition and sustainable development issues are today a reality that cannot be ignored given the negative impacts of human activities on their environments. On the other side, an increasingly important digitization of these environments results in the generation of massive volumes of digital traces, which are all signs of actors’ activities. A significant challenge is to understand the ins and outs of environmental impact due activities and considering Emprise of Energy (EmE) as a key indicator and how this indicator can strongly change from an activity to another. Our approach considers the identification of Practice on the basis of these digital traces generated by human and non-human entities during specific activities. Practice (instantiation of activity) uses more or less resources (physical and virtual) during their existence. Be able to identify which one is more resources dependent would help to better understand how to promote ecological transition. Promoting or at least identifying on the basis of quantifiable indicators (i.e Energy Emprise), practices that have a low impact on the environment, could be an innovative approach. These practices, in the sense of coordination of multiple heterogeneous entities in time and space, can be formalized in the form of multidimensional structures activities - Hypergraph of Activities – using the theory of Assemblage (Agencement in french) and using a set of mathematical tool (Simplicial Complexes, Hypernetworks). This research attempts to model the phenomenon of human and not human activity based on the characterization of the context (massive contextual data). These Assemblages are calculated and represented in an research application (IMhoTEP) which aims to build these complex structures not based on a priori entities’ classification, but by focusing on the relationships that they maintain in several dimensions. The main goal is to offer a decision tool which support actors’ ecological transition by understand activities inducing consumption or production of resources. These academic research in the field of computer science is based continuous digitization of physical and virtual spaces, particularly highly connected urban areas (Smart City, Internet of Everything).