Structural and Spectral Properties of Chordal Ring, Multi-Ring, and Mixed Graphs

The chordal ring (CR) graphs are a well-known family of graphs used to model some interconnection networks for computer systems in which all nodes are in a cycle. Generalizing the CR graphs, in this paper, we introduce the families of chordal multi-ring (CMR), chordal ring mixed (CRM), and chordal m...

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Autores: Reyes, Mónica Andrea, Dalfó, Cristina, Fiol Mora, Miguel Ángel
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2024
País:España
Institución:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repositorio:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat:10459.1/466693
Acceso en línea:https://doi.org/10.3390/sym16091135
https://hdl.handle.net/10459.1/466693
Access Level:acceso abierto
Palabra clave:Chordal ring graphs
Diameter
The degree/diameter problem
Lift graphs
Abelian group
Plane tessellations
Polynomial matrix
Spectrum
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spelling Structural and Spectral Properties of Chordal Ring, Multi-Ring, and Mixed GraphsReyes, Mónica AndreaDalfó, CristinaFiol Mora, Miguel ÁngelChordal ring graphsDiameterThe degree/diameter problemLift graphsAbelian groupPlane tessellationsPolynomial matrixSpectrumThe chordal ring (CR) graphs are a well-known family of graphs used to model some interconnection networks for computer systems in which all nodes are in a cycle. Generalizing the CR graphs, in this paper, we introduce the families of chordal multi-ring (CMR), chordal ring mixed (CRM), and chordal multi-ring mixed (CMRM) graphs. In the case of mixed graphs, we can have edges (without direction) and arcs (with direction). The chordal ring and chordal ring mixed graphs are bipartite and 3-regular. They consist of a number r (for ≥1) of (undirected or directed) cycles with some edges (the chords) joining them. In particular, for CMR, when =1, that is, with only one undirected cycle, we obtain the known families of chordal ring graphs. Here, we used plane tessellations to represent our chordal multi-ring graphs. This allowed us to obtain their maximum number of vertices for every given diameter. Additionally, we computationally obtained their minimum diameter for any value of the number of vertices. Moreover, when seen as a lift graph (also called voltage graph) of a base graph on Abelian groups, we obtained closed formulas for the spectrum, that is, the eigenvalue multi-set of its adjacency matrix.This research has been funded by AGAUR from the Catalan Government under project 2021SGR00434 and MICINN from the Spanish Government under project PID2020-115442RB-I00. M. A. Fiol’s research was also supported by a grant from the Universitat Politècnica de Catalunya with references AGRUPS-2022 and AGRUPS-2023.MDPI2024info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttps://doi.org/10.3390/sym16091135https://hdl.handle.net/10459.1/466693reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)Inglésinfo:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PID2020-115442RB-I00Reproducció del document publicat a https://doi.org/10.3390/sym16091135Symmetry, 2024, vol. 16, p.1135cc-by, (c) Reyes et al., 2024Attribution 4.0 Internationalinfo:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by/4.0/oai:recercat.cat:10459.1/4666932026-05-29T05:05:01Z
dc.title.none.fl_str_mv Structural and Spectral Properties of Chordal Ring, Multi-Ring, and Mixed Graphs
title Structural and Spectral Properties of Chordal Ring, Multi-Ring, and Mixed Graphs
spellingShingle Structural and Spectral Properties of Chordal Ring, Multi-Ring, and Mixed Graphs
Reyes, Mónica Andrea
Chordal ring graphs
Diameter
The degree/diameter problem
Lift graphs
Abelian group
Plane tessellations
Polynomial matrix
Spectrum
title_short Structural and Spectral Properties of Chordal Ring, Multi-Ring, and Mixed Graphs
title_full Structural and Spectral Properties of Chordal Ring, Multi-Ring, and Mixed Graphs
title_fullStr Structural and Spectral Properties of Chordal Ring, Multi-Ring, and Mixed Graphs
title_full_unstemmed Structural and Spectral Properties of Chordal Ring, Multi-Ring, and Mixed Graphs
title_sort Structural and Spectral Properties of Chordal Ring, Multi-Ring, and Mixed Graphs
dc.creator.none.fl_str_mv Reyes, Mónica Andrea
Dalfó, Cristina
Fiol Mora, Miguel Ángel
author Reyes, Mónica Andrea
author_facet Reyes, Mónica Andrea
Dalfó, Cristina
Fiol Mora, Miguel Ángel
author_role author
author2 Dalfó, Cristina
Fiol Mora, Miguel Ángel
author2_role author
author
dc.subject.none.fl_str_mv Chordal ring graphs
Diameter
The degree/diameter problem
Lift graphs
Abelian group
Plane tessellations
Polynomial matrix
Spectrum
topic Chordal ring graphs
Diameter
The degree/diameter problem
Lift graphs
Abelian group
Plane tessellations
Polynomial matrix
Spectrum
description The chordal ring (CR) graphs are a well-known family of graphs used to model some interconnection networks for computer systems in which all nodes are in a cycle. Generalizing the CR graphs, in this paper, we introduce the families of chordal multi-ring (CMR), chordal ring mixed (CRM), and chordal multi-ring mixed (CMRM) graphs. In the case of mixed graphs, we can have edges (without direction) and arcs (with direction). The chordal ring and chordal ring mixed graphs are bipartite and 3-regular. They consist of a number r (for ≥1) of (undirected or directed) cycles with some edges (the chords) joining them. In particular, for CMR, when =1, that is, with only one undirected cycle, we obtain the known families of chordal ring graphs. Here, we used plane tessellations to represent our chordal multi-ring graphs. This allowed us to obtain their maximum number of vertices for every given diameter. Additionally, we computationally obtained their minimum diameter for any value of the number of vertices. Moreover, when seen as a lift graph (also called voltage graph) of a base graph on Abelian groups, we obtained closed formulas for the spectrum, that is, the eigenvalue multi-set of its adjacency matrix.
publishDate 2024
dc.date.none.fl_str_mv 2024
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://doi.org/10.3390/sym16091135
https://hdl.handle.net/10459.1/466693
url https://doi.org/10.3390/sym16091135
https://hdl.handle.net/10459.1/466693
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PID2020-115442RB-I00
Reproducció del document publicat a https://doi.org/10.3390/sym16091135
Symmetry, 2024, vol. 16, p.1135
dc.rights.none.fl_str_mv cc-by, (c) Reyes et al., 2024
Attribution 4.0 International
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by/4.0/
rights_invalid_str_mv cc-by, (c) Reyes et al., 2024
Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.publisher.none.fl_str_mv MDPI
publisher.none.fl_str_mv MDPI
dc.source.none.fl_str_mv reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
repository.name.fl_str_mv
repository.mail.fl_str_mv
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