Distance matrices on the H-join of graphs: A general result and applications

Detalhes bibliográficos
Autor(a) principal: Cardoso, Domingos M.
Data de Publicação: 2018
Outros Autores: Díaz, Roberto C., Rojo, Oscar
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
Texto Completo: http://hdl.handle.net/10773/24082
Resumo: Given a graph $H$ with vertices $1,\ldots ,s$ and a set of pairwise vertex disjoint graphs $G_{1},\ldots ,G_{s},$ the vertex $i$ of $H$ is assigned to $G_{i}.$ Let $G$ be the graph obtained from the graphs $G_{1},\ldots ,G_{s}$ and the edges connecting each vertex of $G_{i}$ with all the vertices of $G_{j}$ for all edge $ij$ of $H.$ The graph $G$ is called the $H-join$ of $G_1,\ldots,G_s$. Let $M(G)$ be a matrix on a graph $G$. A general result on the eigenvalues of $M\left( G\right) $, when the all ones vector is an eigenvector of $M\left( G_{i}\right) $ for $i=1,2,\ldots ,s$, is given. This result is applied to obtain the distance eigenvalues, the distance Laplacian eigenvalues and as well as the distance signless Laplacian eigenvalues of $G$ when $G_{1},\ldots ,G_{s}$ are regular graphs. Finally, we introduce the notions of the distance incidence energy and distance Laplacian-energy like of a graph and we derive sharp lower bounds on these two distance energies among all the connected graphs of prescribed order in terms of the vertex connectivity. The graphs for which those bounds are attained are characterized.
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spelling Distance matrices on the H-join of graphs: A general result and applicationsGraph operationsVetex connectivityDistance matrixEigenvaluesDistance incidence energyDistance Laplacian-energy likeGiven a graph $H$ with vertices $1,\ldots ,s$ and a set of pairwise vertex disjoint graphs $G_{1},\ldots ,G_{s},$ the vertex $i$ of $H$ is assigned to $G_{i}.$ Let $G$ be the graph obtained from the graphs $G_{1},\ldots ,G_{s}$ and the edges connecting each vertex of $G_{i}$ with all the vertices of $G_{j}$ for all edge $ij$ of $H.$ The graph $G$ is called the $H-join$ of $G_1,\ldots,G_s$. Let $M(G)$ be a matrix on a graph $G$. A general result on the eigenvalues of $M\left( G\right) $, when the all ones vector is an eigenvector of $M\left( G_{i}\right) $ for $i=1,2,\ldots ,s$, is given. This result is applied to obtain the distance eigenvalues, the distance Laplacian eigenvalues and as well as the distance signless Laplacian eigenvalues of $G$ when $G_{1},\ldots ,G_{s}$ are regular graphs. Finally, we introduce the notions of the distance incidence energy and distance Laplacian-energy like of a graph and we derive sharp lower bounds on these two distance energies among all the connected graphs of prescribed order in terms of the vertex connectivity. The graphs for which those bounds are attained are characterized.Elsevier2020-12-01T00:00:00Z2018-09-07T00:00:00Z2018-09-07info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10773/24082eng0024-379510.1016/j.laa.2018.08.024Cardoso, Domingos M.Díaz, Roberto C.Rojo, Oscarinfo:eu-repo/semantics/embargoedAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2024-05-06T04:17:00Zoai:ria.ua.pt:10773/24082Portal AgregadorONGhttps://www.rcaap.pt/oai/openairemluisa.alvim@gmail.comopendoar:71602024-05-06T04:17Repositório 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 Distance matrices on the H-join of graphs: A general result and applications
title Distance matrices on the H-join of graphs: A general result and applications
spellingShingle Distance matrices on the H-join of graphs: A general result and applications
Cardoso, Domingos M.
Graph operations
Vetex connectivity
Distance matrix
Eigenvalues
Distance incidence energy
Distance Laplacian-energy like
title_short Distance matrices on the H-join of graphs: A general result and applications
title_full Distance matrices on the H-join of graphs: A general result and applications
title_fullStr Distance matrices on the H-join of graphs: A general result and applications
title_full_unstemmed Distance matrices on the H-join of graphs: A general result and applications
title_sort Distance matrices on the H-join of graphs: A general result and applications
author Cardoso, Domingos M.
author_facet Cardoso, Domingos M.
Díaz, Roberto C.
Rojo, Oscar
author_role author
author2 Díaz, Roberto C.
Rojo, Oscar
author2_role author
author
dc.contributor.author.fl_str_mv Cardoso, Domingos M.
Díaz, Roberto C.
Rojo, Oscar
dc.subject.por.fl_str_mv Graph operations
Vetex connectivity
Distance matrix
Eigenvalues
Distance incidence energy
Distance Laplacian-energy like
topic Graph operations
Vetex connectivity
Distance matrix
Eigenvalues
Distance incidence energy
Distance Laplacian-energy like
description Given a graph $H$ with vertices $1,\ldots ,s$ and a set of pairwise vertex disjoint graphs $G_{1},\ldots ,G_{s},$ the vertex $i$ of $H$ is assigned to $G_{i}.$ Let $G$ be the graph obtained from the graphs $G_{1},\ldots ,G_{s}$ and the edges connecting each vertex of $G_{i}$ with all the vertices of $G_{j}$ for all edge $ij$ of $H.$ The graph $G$ is called the $H-join$ of $G_1,\ldots,G_s$. Let $M(G)$ be a matrix on a graph $G$. A general result on the eigenvalues of $M\left( G\right) $, when the all ones vector is an eigenvector of $M\left( G_{i}\right) $ for $i=1,2,\ldots ,s$, is given. This result is applied to obtain the distance eigenvalues, the distance Laplacian eigenvalues and as well as the distance signless Laplacian eigenvalues of $G$ when $G_{1},\ldots ,G_{s}$ are regular graphs. Finally, we introduce the notions of the distance incidence energy and distance Laplacian-energy like of a graph and we derive sharp lower bounds on these two distance energies among all the connected graphs of prescribed order in terms of the vertex connectivity. The graphs for which those bounds are attained are characterized.
publishDate 2018
dc.date.none.fl_str_mv 2018-09-07T00:00:00Z
2018-09-07
2020-12-01T00:00:00Z
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/10773/24082
url http://hdl.handle.net/10773/24082
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv 0024-3795
10.1016/j.laa.2018.08.024
dc.rights.driver.fl_str_mv info:eu-repo/semantics/embargoedAccess
eu_rights_str_mv embargoedAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
instacron:RCAAP
instname_str Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
instacron_str 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
repository.mail.fl_str_mv mluisa.alvim@gmail.com
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