Publicação
An introduction to Graphs on Groups
| 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 |
| _version_ | 1865920839830994944 |
|---|---|
| author | Andrade, Maria Isabel Diaz |
| author_facet | Andrade, Maria Isabel Diaz Andrade, Maria Isabel Diaz |
| author_role | author |
| contributor_name_str_mv | Torres, Maria Manuel Correia Ferreira, Jorge Nélio Marques Repositório Científico de Acesso Aberto da ULisboa |
| country_str | PT |
| creators_json_str | [{\"Person.name\":\"Andrade, Maria Isabel Diaz\"}] |
| datacite.contributors.contributor.contributorName.fl_str_mv | Torres, Maria Manuel Correia Ferreira, Jorge Nélio Marques Repositório Científico de Acesso Aberto da ULisboa |
| datacite.creators.creator.creatorName.fl_str_mv | Andrade, Maria Isabel Diaz |
| datacite.date.Accepted.fl_str_mv | 2024-01-01T00:00:00Z |
| datacite.date.available.fl_str_mv | 2025-01-08T17:19:43Z |
| datacite.date.embargoed.fl_str_mv | 2025-01-08T17:19:43Z |
| datacite.rights.fl_str_mv | http://purl.org/coar/access_right/c_abf2 |
| datacite.subjects.subject.fl_str_mv | Grafos em grupos Hierarquia de grafos Diferença na hierarquia Conectividade Diâmetro Teses de mestrado - 2024 |
| datacite.titles.title.fl_str_mv | An introduction to Graphs on Groups |
| dc.contributor.none.fl_str_mv | Torres, Maria Manuel Correia Ferreira, Jorge Nélio Marques Repositório Científico de Acesso Aberto da ULisboa |
| dc.creator.none.fl_str_mv | Andrade, Maria Isabel Diaz |
| dc.date.Accepted.fl_str_mv | 2024-01-01T00:00:00Z |
| dc.date.available.fl_str_mv | 2025-01-08T17:19:43Z |
| dc.date.embargoed.fl_str_mv | 2025-01-08T17:19:43Z |
| dc.format.none.fl_str_mv | application/pdf |
| dc.identifier.none.fl_str_mv | http://hdl.handle.net/10400.5/97009 |
| dc.language.none.fl_str_mv | eng |
| dc.rights.none.fl_str_mv | http://purl.org/coar/access_right/c_abf2 |
| dc.subject.none.fl_str_mv | Grafos em grupos Hierarquia de grafos Diferença na hierarquia Conectividade Diâmetro Teses de mestrado - 2024 |
| dc.title.fl_str_mv | An introduction to Graphs on Groups |
| dc.type.none.fl_str_mv | http://purl.org/coar/resource_type/c_bdcc |
| description | 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. |
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| format | masterThesis |
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| id | ul_d3c2c6d9500a08f74b9e9e1db252283c |
| identifier.url.fl_str_mv | http://hdl.handle.net/10400.5/97009 |
| instacron_str | ul |
| institution | Universidade de Lisboa |
| instname_str | Universidade de Lisboa |
| language | eng |
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| network_name_str | Repositório da Universidade de Lisboa |
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| organization_str_mv | urn:organizationAcronym:ul |
| person_str_mv | Andrade, Maria Isabel Diaz |
| publishDate | 2024 |
| reponame_str | Repositório da Universidade de Lisboa |
| repository_id_str | urn:repositoryAcronym:ul |
| service_str_mv | urn:repositoryAcronym:ul |
| spelling | engpt_PTThe 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.application/pdfpt_PTAn introduction to Graphs on GroupsAndrade, Maria Isabel DiazTorres, Maria Manuel CorreiaFerreira, Jorge Nélio MarquesHostingInstitutionOrganizationalRepositório Científico de Acesso Aberto da ULisboae-mailmailto:repositorio@reitoria.ulisboa.ptrepositorio@reitoria.ulisboa.ptURNurn:tid:2038789652025-01-08T17:19:43Z202420242024-01-01T00:00:00ZHandlehttp://hdl.handle.net/10400.5/97009http://purl.org/coar/access_right/c_abf2open accessGrafos em gruposHierarquia de grafosDiferença na hierarquiaConectividadeDiâmetroTeses de mestrado - 2024509229 bytesliteraturehttp://purl.org/coar/resource_type/c_bdccmaster thesishttp://purl.org/coar/access_right/c_abf2application/pdffulltexthttps://repositorio.ulisboa.pt/bitstreams/04785c9b-ddbb-409c-aa03-750109e80b37/download |
| spellingShingle | An introduction to Graphs on Groups An introduction to Graphs on Groups Andrade, Maria Isabel Diaz Grafos em grupos Hierarquia de grafos Diferença na hierarquia Conectividade Diâmetro Teses de mestrado - 2024 Andrade, Maria Isabel Diaz Grafos em grupos Hierarquia de grafos Diferença na hierarquia Conectividade Diâmetro Teses de mestrado - 2024 |
| status | SINGLETON |
| subject.fl_str_mv | Grafos em grupos Hierarquia de grafos Diferença na hierarquia Conectividade Diâmetro Teses de mestrado - 2024 |
| title | An introduction to Graphs on Groups |
| title_full | An introduction to Graphs on Groups |
| title_fullStr | An introduction to Graphs on Groups An introduction to Graphs on Groups |
| title_full_unstemmed | An introduction to Graphs on Groups An introduction to Graphs on Groups |
| title_short | An introduction to Graphs on Groups |
| title_sort | An introduction to Graphs on Groups |
| topic | Grafos em grupos Hierarquia de grafos Diferença na hierarquia Conectividade Diâmetro Teses de mestrado - 2024 |
| topic_facet | Grafos em grupos Hierarquia de grafos Diferença na hierarquia Conectividade Diâmetro Teses de mestrado - 2024 |
| url | http://hdl.handle.net/10400.5/97009 |
| visible | 1 |