Necessary and sufficient conditions for a Hamiltonian graph
Autor(a) principal: | |
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Data de Publicação: | 2012 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10773/13474 |
Resumo: | A graph is singular if the zero eigenvalue is in the spectrum of its 0-1 adjacency matrix A. If an eigenvector belonging to the zero eigenspace of A has no zero entries, then the singular graph is said to be a core graph. A ( k,t)-regular set is a subset of the vertices inducing a k -regular subgraph such that every vertex not in the subset has t neighbours in it. We consider the case when k=t which relates to the eigenvalue zero under certain conditions. We show that if a regular graph has a ( k,k )-regular set, then it is a core graph. By considering the walk matrix we develop an algorithm to extract ( k,k )-regular sets and formulate a necessary and sufficient condition for a graph to be Hamiltonian. |
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Necessary and sufficient conditions for a Hamiltonian graphHamiltonian graphsSingular graphsGraph eigenvaluesA graph is singular if the zero eigenvalue is in the spectrum of its 0-1 adjacency matrix A. If an eigenvector belonging to the zero eigenspace of A has no zero entries, then the singular graph is said to be a core graph. A ( k,t)-regular set is a subset of the vertices inducing a k -regular subgraph such that every vertex not in the subset has t neighbours in it. We consider the case when k=t which relates to the eigenvalue zero under certain conditions. We show that if a regular graph has a ( k,k )-regular set, then it is a core graph. By considering the walk matrix we develop an algorithm to extract ( k,k )-regular sets and formulate a necessary and sufficient condition for a graph to be Hamiltonian.Charles Babbage Research Centre2015-02-24T12:53:44Z2012-02-01T00:00:00Z2012-02info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10773/13474eng0835-3026Sciriha, ICardoso, Domingos M.info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2024-02-22T11:24:28Zoai:ria.ua.pt:10773/13474Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T02:49:18.280344Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
Necessary and sufficient conditions for a Hamiltonian graph |
title |
Necessary and sufficient conditions for a Hamiltonian graph |
spellingShingle |
Necessary and sufficient conditions for a Hamiltonian graph Sciriha, I Hamiltonian graphs Singular graphs Graph eigenvalues |
title_short |
Necessary and sufficient conditions for a Hamiltonian graph |
title_full |
Necessary and sufficient conditions for a Hamiltonian graph |
title_fullStr |
Necessary and sufficient conditions for a Hamiltonian graph |
title_full_unstemmed |
Necessary and sufficient conditions for a Hamiltonian graph |
title_sort |
Necessary and sufficient conditions for a Hamiltonian graph |
author |
Sciriha, I |
author_facet |
Sciriha, I Cardoso, Domingos M. |
author_role |
author |
author2 |
Cardoso, Domingos M. |
author2_role |
author |
dc.contributor.author.fl_str_mv |
Sciriha, I Cardoso, Domingos M. |
dc.subject.por.fl_str_mv |
Hamiltonian graphs Singular graphs Graph eigenvalues |
topic |
Hamiltonian graphs Singular graphs Graph eigenvalues |
description |
A graph is singular if the zero eigenvalue is in the spectrum of its 0-1 adjacency matrix A. If an eigenvector belonging to the zero eigenspace of A has no zero entries, then the singular graph is said to be a core graph. A ( k,t)-regular set is a subset of the vertices inducing a k -regular subgraph such that every vertex not in the subset has t neighbours in it. We consider the case when k=t which relates to the eigenvalue zero under certain conditions. We show that if a regular graph has a ( k,k )-regular set, then it is a core graph. By considering the walk matrix we develop an algorithm to extract ( k,k )-regular sets and formulate a necessary and sufficient condition for a graph to be Hamiltonian. |
publishDate |
2012 |
dc.date.none.fl_str_mv |
2012-02-01T00:00:00Z 2012-02 2015-02-24T12:53:44Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10773/13474 |
url |
http://hdl.handle.net/10773/13474 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
0835-3026 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
Charles Babbage Research Centre |
publisher.none.fl_str_mv |
Charles Babbage Research Centre |
dc.source.none.fl_str_mv |
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Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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