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An introduction to Graphs on Groups

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Resumo:The present dissertation, titled An Introduction to Graphs on Groups, aims to study connections between two fundamental areas of mathematics: groups and graphs. In the first chapter we introduce the basic definitions and results that will be needed throughout this essay. It has four sections. The first one is dedicated to Graph Theory, and it is mainly composed by basic definitions and examples. The rest of the chapter focuses on Group Theory. In this part of the essay, we introduce the different classes of groups with which we will be working on. Chapter 2 is divided into four sections, one for each graph on group we will be working on: the commuting graph, the power graph, the enhanced power graph and the non-generating graph. Here we study for which groups each of these graphs is complete or null, we deduce some results regarding maximal cliques and present a few results regarding diameter and connectivity. Several examples are given. The next chapter has as objective to study some differences (in the sense of subtraction) between the graphs presented in Chapter 2. Hence, we begin Chapter 3 with the definition of graph hierarchy and we show that it is well defined. Next, we study when two graphs in said hierarchy are equal. We then compute the difference between pairs of graphs in the hierarchy. In particular, we study the difference between the enhanced power graph and the power graph and the difference between the non-generating graph and the commuting graph. For each of these new classes of graphs, we determine their isolated vertices, study connectivity, diameter, and give several examples. We finish the main body of this dissertation with Chapter 4. Here we define and study two differences of graphs that have not yet, to the best of our knowledge, been considered.
Autores principais:Andrade, Maria Isabel Diaz
Assunto:Grafos em grupos Hierarquia de grafos Diferença na hierarquia Conectividade Diâmetro Teses de mestrado - 2024
Ano:2024
País:Portugal
Tipo de documento:dissertação de mestrado
Tipo de acesso:acesso aberto
Instituição associada:Universidade de Lisboa
Idioma:inglês
Origem:Repositório da Universidade de Lisboa

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