[Abstract] This thesis is divided into two parts, both containing algorithms for dealing with matrices and matrix groups. Part I is concerned with individual matrices over an arbitrary field. Our algorithms make use of a sequence called the rank profile which is related to the linear dependence relations between the columns of a matrix. First we look at LSP decompositions of matrices as defined by Ibarra et al. in 1982. This decomposition is related to, and a little more general than, the LUP decomposition. The algorithm given by Ibarra et al. to compute an LSP decomposition was only defined for m?n matrices where m ≤ n and is claimed to have the same asymptotic cost as matrix multiplication. We prove that their cost analysis overlooked some aspects of the computation and present a new version of the algorithm which finds both an LSP decomposition and the rank profile of any matrix. The cost of our algorithm is the same as that claimed by Ibarra et al. when m ≤ n and has a similar cost when m > n. One of the steps in the Ibarra et al. algorithm is not completely explicit, so that any one of several choices can be made. Our algorithm is designed so that the particular choice made at this point allows for the simultaneous calculation of the rank profile. Next we study algorithms to find the characteristic polynomial of a square matrix. The current fastest algorithm to find the characteristic polynomial of a square matrix was developed by Keller-Gehrig in 1985. We present a new, simpler version of this algorithm with the same cost which makes the algorithm?s reliance on the rank profile explicit. In Part II we present generalised sifting, a scheme for creating Monte Carlo black box constructive group recognition algorithms. Generalised sifting is designed to facilitate computation in a known group, specifically re-writing arbitrary elements as words or straight-line programs in a standard generating set. It can also be used to create membership tests in black-box groups. Generalised sifting was inspired by the subgroup sifting techniques originally introduced by Sims in 1970 but uses a chain of subsets rather than subgroups. We break the problem down into a sequence of separately analysed and proven steps which sift down into each subset in turn ... All of the algorithms in Parts I and II are given with a theoretical proof and (where appropriate) complexity analysis. The LSP decomposition, characteristic polynomial and generalised sifting algorithms have all been implemented and tested in the computer algebra package GAP.

Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2017-01-26T09:20:52Z No. of bitstreams: 2 Dissertação - Jânio Alves dos Santos Júnior - 2016.pdf: 3798217 bytes, checksum: c2acd93260ead52c126f4b37d994825f (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-01-30T10:25:56Z (GMT) No. of bitstreams: 2 Dissertação - Jânio Alves dos Santos Júnior - 2016.pdf: 3798217 bytes, checksum: c2acd93260ead52c126f4b37d994825f (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-01-30T10:25:56Z (GMT). No. of bitstreams: 2 Dissertação - Jânio Alves dos Santos Júnior - 2016.pdf: 3798217 bytes, checksum: c2acd93260ead52c126f4b37d994825f (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-12-14 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work aims to study some topics of graph theory in order to solve some problems. In order to complement, we approached a light study of matrices, eigenvalues and eigenvectors. The first problem is known as Königsberg Bridge Problem, where this was considered the problem that gave rise to the study on graphs. The House Problem is a joke, which shows us several propositions about planar and bipartite graphs. Some models we can relate graphs, such as we can observe in the problem of cannibals and in the game of chess. Finally, we will work with applications in the adjacency matrix as in the Problem of the Condominium of Farms and in the Number of Possible Paths in a graph, where we will work with geometric figures, apparently resolving a counting problem using eigenvalues and graph. As a methodological support will be approached Linear Algebra. / O objetivo deste trabalho é estudar alguns tópicos da teoria de grafos com o intuito de resolver alguns problemas. Para complementar, abordamos um leve estudo de matrizes, autovalores e autovetores. O primeiro problema é conhecido como o Problema da Ponte de Königsberg, onde tal, foi considerado o problema que deu origem ao estudo sobre grafos. O Problema das Casas que é uma brincadeira, que nos mostra várias proposições sobre grafos planares e bipartidos. Alguns modelos que podemos relacionar grafos, tais como veremos no problema dos canibais e no jogo de xadrez. Por fim, trabalharemos com aplicações na matriz de adjacência como no problema do Condomínio de Chácaras e no Número de Caminhos Possíveis em um Grafo, onde trabalharemos com figuras geométricas, resolvendo aparentemente um problema de contagem, utilizando autovalores e grafos. Como suporte metodológico será abordado Álgebra Linear.

El principal objectiu d'aquesta tesi és el de contribuir a l'estudi de l'existència i classificació dels grafs i digrafs que puguin admetre el màxim nombre de vèrtexs sota determinades condicions donats el grau i el diàmetre. Aquest estudi consta de tres parts ben diferenciades, una sobre digrafs i dos sobre grafs. En el treball relacionat amb els digrafs demostrem que els digrafs quasi de Moore de diàmetre k = 3 i qualsevol grau no existeixen. Així mateix provem la no existència dels digrafs quasi de Moore de diàmetre 4 i qualsevol grau assumint la irreductibilitat en Q[x] de certs polinomis. En quan als grafs ens hem centrat en l'existència dels de grau d, diàmetre 2 i defecte 2, anomenats (d,2,2)-grafs i assumint la irreductibilitat en Q[x] de certs polinomis provem que no existeixen per a cap grau. A més provem que no existeixen per a graus entre 4 i 50. Finalment estudiem els grafs radials de Moore de grau d i radi k. Proposem diferents mesures per classificar-los d'acord a la proximitat de les seves propietats a les d'un graf de Moore i ordenem segons aquestes mesures tots els grafs radials de Moore en els casos (d,k) = {(3,2), (3,3), (4,2)}. / El principal objetivo de esta tesis es el de contribuir al estudio de la existencia y clasificación de los grafos y digrafos que puedan admitir el máximo número de vértices bajo determinadas condiciones dados el grado y el diámetro. Este estudio consta de tres partes bien diferenciadas, una sobre digrafos y dos sobre grafos. En el trabajo relacionado con los digrafos demostramos que los digrafos casi de Moore de diámetro k = 3 y cualquier grado no existen. Asimismo probamos la no existencia de los digrafos casi de Moore de diámetro 4 y cualquier grado suponiendo la irreducibilidad en Q[x] de ciertos polinomios. En cuanto a los grafos nos hemos centrado en la existencia de los de grado d, diámetro 2 y defecto 2, llamados (d,2,2)-grafos y suponiendo la irreducibilidad en Q[x] de ciertos polinomios probamos que no existen para ningún grado. Además probamos que no existen para grados entre 4 y 50. Finalmente estudiamos los grafos radiales de Moore de grado d y radio k. Proponemos diferentes medidas para clasificarlos de acuerdo a la proximidad de sus propiedades a las de un grafo de Moore y ordenamos según estas medidas todos los grafos radiales de Moore en los casos (d, k) = {(3,2), (3,3), (4,2)}. / The main goal of this thesis is to contribute to the study of the existence and classification of graphs and digraphs that can achieve the maximum number of vertices under certain conditions given the degree and the diameter. This study consists of three differenciated parts, one on digraphs and two on graphs. The work on digraphs focuses on almost Moore digraphs. We prove that they do not exist for diameter 3 and any degree. Besides, we prove the non-existence of almost Moore digraphs of diameter 4 assuming the irreducibility in Q[x] of certain polynomials. Concerning graphs, we discuss the existence of graphs of degree d, diameter 2 and defect 2. Assuming the irreducibility in Q[x] of certain polynomials we prove their non existence. We also show they do not exist for degrees between 4 and 50. Finally we study radial Moore graphs of degree d and radius k. We propose different measures for classifying them in terms of their proximity to extremal properties of a Moore graph. By means of our measures, we are able to enumerate all radial Moore graphs for the cases (d, k) = {(3.2), (3.3), (4.2)}.

[Truncated abstract] Online games are an exciting new trend in the consumption of entertainment and provide the opportunity to examine selected antecedents of online game-playing based on studying the cognitive, experiential and habitual factors. This study was divided into two parts. The first part analysed the structural relations among research variables that might explain online game-playing using the Structural Equation Modeling (SEM) techniques. These analyses were conducted on a final sample of 218 online gamers. Specific issues examined were: If the variables of Perceived Game Performance, Satisfaction, Hedonic Responses, Flow and Habit Strength influence the Intention to Replay an online game. The importance of factors such as Hedonic Responses and Flow on Satisfaction in online game play. In addition to the SEM, analyses of the participants? reported past playing behaviour were conducted to test whether past game play was simply a matter of random frequency of past behaviour, or followed the specific pattern of the Negative Binomial Distribution (NBD). … The playing-time distribution was not significantly different to the Gamma distribution, in which the largest number of gamers plays for a short time (light gamers) and only a few gamers account for a large proportion of playing time (heavy gamers). Therefore, the reported time play followed a simple and predictable NBD pattern (Chisquare=. 390; p>.05). This study contributes to knowledge in the immediate field of online games and to the wider body of literature on consumer research. The findings demonstrate that gamers tend to act habitually in their playing behaviour. These findings support the argument that past behaviour (habit) is a better explanation of future behaviour than possible cognitive and affective explanations, especially for the apparent routinesed behaviour pattern on online games. The pattern of online game-playing is consistent with the finding of the NBD pattern in television viewing, in which the generalisability of the NBD model has been found in stable environments of repetitive behaviour. This supports the application of the NBD to areas beyond those of patterns in gambling and the purchase of consumer items. The findings have implications both for managerial and public policy decision-making.