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Introdução à teoria espectral de grafos
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| 1.77 MB | Adobe PDF |
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Abstract(s)
O objetivo principal desta dissertação é fazer uma introdução à teoria espectral de grafos. Abordaremos algumas propriedades dos grafos, nomeadamente o polinómio característico, os valores e vetores próprios de matrizes associadas aos grafos. Nesta dissertação iremos dar mais relevância à matriz de adjacência e à matriz laplaciana, fazendo uma análise de alguns tipos especí…cos de grafos e dos respetivos espectros. Iremos estudar o conceito de energia de um grafo e de medidas de centralidade associadas aos grafos e aos seus conceitos inerentes. Serão apresentadas duas aplicações no decorrer da dissertação, uma referente à Química e ao carbono quaternário, com o objetivo de descobrir se o carbono quaternário existe ou não na molécula em estudo e, outra referente às medidas de centralidade em que será feito a análise do jogo de futebol “Portugal vs França” a contar para a …nal do Europeu de 2016, com o intuito de descobrir as performances dos jogadores.
The main purpose of this dissertation is to make an introduction to the spectral graph theory. We will study some properties of graphs, namely the characteristic polynomial, the eigenvalues and the eigenvectors speci…c to matrices associated with graphs. In this dissertation, we will give more importance to the adjacency matrix and to the Laplacian matrix and we will do an analysis of some speci…c types of graphs and their respective spectra. We will study the energy of a graph and the measures of centrality associated to graphs and their inherent concepts. Two practical applications will be presented throughout the dissertation, one related to chemistry and quaternary carbon, in order to …nd out whether or not the quaternary carbon exists in the molecule under study, and another one related to the centrality measures in which the analysis of the football match “Portugal versus France”, that de…ned the European Champion of 2016 will be done, with the objective of sorting out the performances of the players on the …eld.
The main purpose of this dissertation is to make an introduction to the spectral graph theory. We will study some properties of graphs, namely the characteristic polynomial, the eigenvalues and the eigenvectors speci…c to matrices associated with graphs. In this dissertation, we will give more importance to the adjacency matrix and to the Laplacian matrix and we will do an analysis of some speci…c types of graphs and their respective spectra. We will study the energy of a graph and the measures of centrality associated to graphs and their inherent concepts. Two practical applications will be presented throughout the dissertation, one related to chemistry and quaternary carbon, in order to …nd out whether or not the quaternary carbon exists in the molecule under study, and another one related to the centrality measures in which the analysis of the football match “Portugal versus France”, that de…ned the European Champion of 2016 will be done, with the objective of sorting out the performances of the players on the …eld.
Description
Keywords
Energia de um grafo Espectro de um grafo Grafo Matriz de adjacência Matriz laplaciana Medidas de centralidade Adjacency matrix Centrality measures Energy of a graph Graph Laplacian matrix Spectrum of a graph Matemática . Faculdade de Ciências Exatas e da Engenharia