Loading...
Publication
Triângulo DNG
| Name: | Description: | Size: | Format: | |
|---|---|---|---|---|
| 5.45 MB | Adobe PDF |
Authors
Advisor(s)
Abstract(s)
Com o estudo da teoria dos grafos e da teoria dos designs surgiu o principal ob jetivo desta dissertação, nomeadamente, a construção do triângulo DNG, ou seja, a
construção de ligações entre a teoria dos designs combinatórios, a teoria dos números
e a teoria dos grafos.
Nesta dissertação, em primeiro lugar, fez-se uma introdução teórica com alguns
exemplos, de cada uma das teorias pertencentes ao triângulo DNG e, de seguida,
demonstrou-se como estas poderiam estar ligadas, aplicando-as em diversos proble mas.
Dentro do grupo dos designs, foi dada a devida relevância aos BIBDs (Balanced
Incomplete Block Designs), em que conseguimos calcular a matriz de Hadamard
para alguns BIBDs não simétricos.
Em relação à teoria dos números, mantivemos o nosso foco nos números euleri anos, por percebermos que existia uma relação entre estes e os BIBDs.
Na teoria dos grafos, para além dos grafos completos, importantíssimos na relação
da teoria dos grafos com a teoria dos designs, demos importância ao polinómio de
Tutte e a sua relação com os grafos bipartidos.
As aplicações do triângulo DNG em situações recorrentes da nossa vida servem
para provar que a matemática está interligada e sempre presente, mesmo que por
vezes a sua importância não seja visível para todos.
With the study of the graph theory and the design theory, the main objective of this dissertation emerged, namely, on the construction of the DNG triangle, that is, the construction of connections between the theory of combinatorial designs, the theory of numbers and the theory of graphs. In this dissertation, in first place, a theoretical introduction was made with some examples, of each of the theories belonging to the DNG triangle, and then, it was demonstrated how they could be linked, applying them in several problems. Within the design group, due importance was given to BIBDs (Balanced Incom plete Block Designs), in which we were able to calculate the Hadamard matrix for some non-symmetric BIBDs. Regarding number theory, we kept our focus on Eulerian numbers, as we realized that there was a relationship between them and the BIBDs. In graph theory, in addition to the complete graphs, that are very important in the relation between graph theory and design theory, we gave importance to Tutte’s polynomial and its relation to bipartite graphs. The applications of the DNG triangle in recurring situations of our lives proves that mathematics is interconnected and always present, even though its importance is not visible to everyone.
With the study of the graph theory and the design theory, the main objective of this dissertation emerged, namely, on the construction of the DNG triangle, that is, the construction of connections between the theory of combinatorial designs, the theory of numbers and the theory of graphs. In this dissertation, in first place, a theoretical introduction was made with some examples, of each of the theories belonging to the DNG triangle, and then, it was demonstrated how they could be linked, applying them in several problems. Within the design group, due importance was given to BIBDs (Balanced Incom plete Block Designs), in which we were able to calculate the Hadamard matrix for some non-symmetric BIBDs. Regarding number theory, we kept our focus on Eulerian numbers, as we realized that there was a relationship between them and the BIBDs. In graph theory, in addition to the complete graphs, that are very important in the relation between graph theory and design theory, we gave importance to Tutte’s polynomial and its relation to bipartite graphs. The applications of the DNG triangle in recurring situations of our lives proves that mathematics is interconnected and always present, even though its importance is not visible to everyone.
Description
Keywords
Grafo Design Número Euleriano Polinómio de Tutte Matriz de Hadamard Graph Eulerian number Tutte polynomial Hadamard matrix Matemática, Estatística e Aplicações . Faculdade de Ciências Exatas e da Engenharia