Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spaces

Autores
Alvarez Hamelin, José Ignacio; Giribet, Juan Ignacio; Mas, Ignacio Agustin; Presenza, Juan Francisco
Año de publicación
2025
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Graph rigidity—the study of vertex realizations in ℝ d and the motions that preserve the induced edge lengths—has been the focus of extensive research for decades. Its equivalency to graph connectivity for d = 1 is well known; thus it can be viewed as a generalization that incorporates geometric constraints. Graph connectivity is commonly quantified by the algebraic connectivity, the second-smallest eigenvalue of the Laplacian matrix. Recently, a graph invariant for quantifying graph rigidity in ℝ d , termed the generalized algebraic connectivity, was introduced. Recognizing the intrinsic relationship between rigidity and connectivity, this article presents new contributions. In particular, we introduce the d -rigidity ratio as a metric for expressing the level of rigidity of a graph in ℝ d relative to its connectivity. We show that this ratio is bounded and provide extremal examples. Additionally, we offer a new upper bound for the generalized algebraic connectivity that depends inversely on the diameter and on the vertex connectivity, thereby improving previous bounds. Moreover, we investigate the relationship between graph rigidity and the diameter—a measure of the graph’s overall extent. We provide the maximal diameter achievable by rigid graphs and show that generalized path graphs serve as extremal examples. Finally, we derive an upper bound for the generalized algebraic connectivity of generalized path graphs that (asymptotically) improves upon existing ones by a factor of four.
Fil: Alvarez Hamelin, José Ignacio. Universidad de Buenos Aires. Facultad de Ingeniería; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Houssay. Instituto de Tecnologías y Ciencias de la Ingeniería "Hilario Fernández Long". Universidad de Buenos Aires. Facultad de Ingeniería. Instituto de Tecnologías y Ciencias de la Ingeniería "Hilario Fernández Long"; Argentina
Fil: Giribet, Juan Ignacio. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Mas, Ignacio Agustin. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Presenza, Juan Francisco. Universidad de Buenos Aires. Facultad de Ingeniería; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Houssay. Instituto de Tecnologías y Ciencias de la Ingeniería "Hilario Fernández Long". Universidad de Buenos Aires. Facultad de Ingeniería. Instituto de Tecnologías y Ciencias de la Ingeniería "Hilario Fernández Long"; Argentina
Materia
graph rigidity
control applications
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/277989
Acceder
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network_name_str CONICET Digital (CONICET)
spelling Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spacesAlvarez Hamelin, José IgnacioGiribet, Juan IgnacioMas, Ignacio AgustinPresenza, Juan Franciscograph rigiditycontrol applicationshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Graph rigidity—the study of vertex realizations in ℝ d and the motions that preserve the induced edge lengths—has been the focus of extensive research for decades. Its equivalency to graph connectivity for d = 1 is well known; thus it can be viewed as a generalization that incorporates geometric constraints. Graph connectivity is commonly quantified by the algebraic connectivity, the second-smallest eigenvalue of the Laplacian matrix. Recently, a graph invariant for quantifying graph rigidity in ℝ d , termed the generalized algebraic connectivity, was introduced. Recognizing the intrinsic relationship between rigidity and connectivity, this article presents new contributions. In particular, we introduce the d -rigidity ratio as a metric for expressing the level of rigidity of a graph in ℝ d relative to its connectivity. We show that this ratio is bounded and provide extremal examples. Additionally, we offer a new upper bound for the generalized algebraic connectivity that depends inversely on the diameter and on the vertex connectivity, thereby improving previous bounds. Moreover, we investigate the relationship between graph rigidity and the diameter—a measure of the graph’s overall extent. We provide the maximal diameter achievable by rigid graphs and show that generalized path graphs serve as extremal examples. Finally, we derive an upper bound for the generalized algebraic connectivity of generalized path graphs that (asymptotically) improves upon existing ones by a factor of four.Fil: Alvarez Hamelin, José Ignacio. Universidad de Buenos Aires. Facultad de Ingeniería; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Houssay. Instituto de Tecnologías y Ciencias de la Ingeniería "Hilario Fernández Long". Universidad de Buenos Aires. Facultad de Ingeniería. Instituto de Tecnologías y Ciencias de la Ingeniería "Hilario Fernández Long"; ArgentinaFil: Giribet, Juan Ignacio. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Mas, Ignacio Agustin. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Presenza, Juan Francisco. Universidad de Buenos Aires. Facultad de Ingeniería; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Houssay. Instituto de Tecnologías y Ciencias de la Ingeniería "Hilario Fernández Long". Universidad de Buenos Aires. Facultad de Ingeniería. Instituto de Tecnologías y Ciencias de la Ingeniería "Hilario Fernández Long"; ArgentinaElsevier Science Inc.2025-12info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/277989Alvarez Hamelin, José Ignacio; Giribet, Juan Ignacio; Mas, Ignacio Agustin; Presenza, Juan Francisco; Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spaces; Elsevier Science Inc.; Linear Algebra and its Applications; 727; 12-2025; 24-360024-3795CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://linkinghub.elsevier.com/retrieve/pii/S0024379525003192info:eu-repo/semantics/altIdentifier/doi/10.1016/j.laa.2025.07.028info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/2505.16015v1info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-12-23T13:37:06Zoai:ri.conicet.gov.ar:11336/277989instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-12-23 13:37:07.12CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spaces
title Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spaces
spellingShingle Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spaces
Alvarez Hamelin, José Ignacio
graph rigidity
control applications
title_short Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spaces
title_full Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spaces
title_fullStr Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spaces
title_full_unstemmed Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spaces
title_sort Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spaces
dc.creator.none.fl_str_mv Alvarez Hamelin, José Ignacio
Giribet, Juan Ignacio
Mas, Ignacio Agustin
Presenza, Juan Francisco
author Alvarez Hamelin, José Ignacio
author_facet Alvarez Hamelin, José Ignacio
Giribet, Juan Ignacio
Mas, Ignacio Agustin
Presenza, Juan Francisco
author_role author
author2 Giribet, Juan Ignacio
Mas, Ignacio Agustin
Presenza, Juan Francisco
author2_role author
author
author
dc.subject.none.fl_str_mv graph rigidity
control applications
topic graph rigidity
control applications
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv Graph rigidity—the study of vertex realizations in ℝ d and the motions that preserve the induced edge lengths—has been the focus of extensive research for decades. Its equivalency to graph connectivity for d = 1 is well known; thus it can be viewed as a generalization that incorporates geometric constraints. Graph connectivity is commonly quantified by the algebraic connectivity, the second-smallest eigenvalue of the Laplacian matrix. Recently, a graph invariant for quantifying graph rigidity in ℝ d , termed the generalized algebraic connectivity, was introduced. Recognizing the intrinsic relationship between rigidity and connectivity, this article presents new contributions. In particular, we introduce the d -rigidity ratio as a metric for expressing the level of rigidity of a graph in ℝ d relative to its connectivity. We show that this ratio is bounded and provide extremal examples. Additionally, we offer a new upper bound for the generalized algebraic connectivity that depends inversely on the diameter and on the vertex connectivity, thereby improving previous bounds. Moreover, we investigate the relationship between graph rigidity and the diameter—a measure of the graph’s overall extent. We provide the maximal diameter achievable by rigid graphs and show that generalized path graphs serve as extremal examples. Finally, we derive an upper bound for the generalized algebraic connectivity of generalized path graphs that (asymptotically) improves upon existing ones by a factor of four.
Fil: Alvarez Hamelin, José Ignacio. Universidad de Buenos Aires. Facultad de Ingeniería; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Houssay. Instituto de Tecnologías y Ciencias de la Ingeniería "Hilario Fernández Long". Universidad de Buenos Aires. Facultad de Ingeniería. Instituto de Tecnologías y Ciencias de la Ingeniería "Hilario Fernández Long"; Argentina
Fil: Giribet, Juan Ignacio. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Mas, Ignacio Agustin. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Presenza, Juan Francisco. Universidad de Buenos Aires. Facultad de Ingeniería; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Houssay. Instituto de Tecnologías y Ciencias de la Ingeniería "Hilario Fernández Long". Universidad de Buenos Aires. Facultad de Ingeniería. Instituto de Tecnologías y Ciencias de la Ingeniería "Hilario Fernández Long"; Argentina
description Graph rigidity—the study of vertex realizations in ℝ d and the motions that preserve the induced edge lengths—has been the focus of extensive research for decades. Its equivalency to graph connectivity for d = 1 is well known; thus it can be viewed as a generalization that incorporates geometric constraints. Graph connectivity is commonly quantified by the algebraic connectivity, the second-smallest eigenvalue of the Laplacian matrix. Recently, a graph invariant for quantifying graph rigidity in ℝ d , termed the generalized algebraic connectivity, was introduced. Recognizing the intrinsic relationship between rigidity and connectivity, this article presents new contributions. In particular, we introduce the d -rigidity ratio as a metric for expressing the level of rigidity of a graph in ℝ d relative to its connectivity. We show that this ratio is bounded and provide extremal examples. Additionally, we offer a new upper bound for the generalized algebraic connectivity that depends inversely on the diameter and on the vertex connectivity, thereby improving previous bounds. Moreover, we investigate the relationship between graph rigidity and the diameter—a measure of the graph’s overall extent. We provide the maximal diameter achievable by rigid graphs and show that generalized path graphs serve as extremal examples. Finally, we derive an upper bound for the generalized algebraic connectivity of generalized path graphs that (asymptotically) improves upon existing ones by a factor of four.
publishDate 2025
dc.date.none.fl_str_mv 2025-12
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/277989
Alvarez Hamelin, José Ignacio; Giribet, Juan Ignacio; Mas, Ignacio Agustin; Presenza, Juan Francisco; Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spaces; Elsevier Science Inc.; Linear Algebra and its Applications; 727; 12-2025; 24-36
0024-3795
CONICET Digital
CONICET
url http://hdl.handle.net/11336/277989
identifier_str_mv Alvarez Hamelin, José Ignacio; Giribet, Juan Ignacio; Mas, Ignacio Agustin; Presenza, Juan Francisco; Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spaces; Elsevier Science Inc.; Linear Algebra and its Applications; 727; 12-2025; 24-36
0024-3795
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://linkinghub.elsevier.com/retrieve/pii/S0024379525003192
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.laa.2025.07.028
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/2505.16015v1
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Elsevier Science Inc.
publisher.none.fl_str_mv Elsevier Science Inc.
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 12.952241

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