On the spectra and spectral radii of token graphs

Let G be a graph on n vertices. The k-token graph (or symmetric k-th power) of G, denoted by Fk(G), has as vertices the (n/k) k-subsets of vertices from G, and two vertices are adjacent when their symmetric difference is a pair of adjacent vertices in G. In particular, Fk(Kn) is the Johnson graph J(...

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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:Universitat de Lleida (UdL)
Repositorio:Repositori Obert UdL
OAI Identifier:oai:repositori.udl.cat:10459.1/464851
Acceso en línea:https://doi.org/10.1007/s40590-023-00583-3
https://hdl.handle.net/10459.1/464851
Access Level:acceso abierto
Palabra clave:Token graph
Adjacency spectrum
Local spectrum
Laplacian spectrum
Algebraic connectivity
Binomial matrix
Spectral radius
Walk-regular graph
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spelling On the spectra and spectral radii of token graphsReyes, Mónica AndreaDalfó, CristinaFiol Mora, Miguel ÁngelToken graphAdjacency spectrumLocal spectrumLaplacian spectrumAlgebraic connectivityBinomial matrixSpectral radiusWalk-regular graphLet G be a graph on n vertices. The k-token graph (or symmetric k-th power) of G, denoted by Fk(G), has as vertices the (n/k) k-subsets of vertices from G, and two vertices are adjacent when their symmetric difference is a pair of adjacent vertices in G. In particular, Fk(Kn) is the Johnson graph J(n, k), which is a distance-regular graph used in coding theory. In this paper, we present some results concerning the (adjacency and Laplacian) spectrum of Fk(G) in terms of the spectrum of G. For instance, when G is walk-regular, an exact value for the spectral radius (or maximum eigenvalue) of Fk(G) is obtained. When G is distance-regular, other eigenvalues of its 2-token graph are derived using the theory of equitable partitions. A generalization of Aldous’ spectral gap conjecture (which is now a theorem) is proposed.The research of C. Dalfó and M. A. Fiol has been partially supported by AGAUR from the Catalan Government under Project 2017SGR1087 and by MICINN from the Spanish Government under Project PGC2018-095471-B-I00. The research of M. A. Fiol was also supported by a grant from the Universitat Politècnica de Catalunya with references AGRUPS-2022 and AGRUPS-2023.Springer2024info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttps://doi.org/10.1007/s40590-023-00583-3https://hdl.handle.net/10459.1/464851reponame:Repositori Obert UdL instname:Universitat de Lleida (UdL)Inglésinfo:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PGC2018-095471-B-I00Reproducció del document publicat a http://doi.org/10.1007/s40590-023-00583-3Boletín de la Sociedad Matemática Mexicana, 2024, vol. 30, art. 11cc-by (c) Reyes et al., 2024Attribution 4.0 Internationalinfo:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by/4.0/oai:repositori.udl.cat:10459.1/4648512026-06-24T12:42:17Z
dc.title.none.fl_str_mv On the spectra and spectral radii of token graphs
title On the spectra and spectral radii of token graphs
spellingShingle On the spectra and spectral radii of token graphs
Reyes, Mónica Andrea
Token graph
Adjacency spectrum
Local spectrum
Laplacian spectrum
Algebraic connectivity
Binomial matrix
Spectral radius
Walk-regular graph
title_short On the spectra and spectral radii of token graphs
title_full On the spectra and spectral radii of token graphs
title_fullStr On the spectra and spectral radii of token graphs
title_full_unstemmed On the spectra and spectral radii of token graphs
title_sort On the spectra and spectral radii of token 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 Token graph
Adjacency spectrum
Local spectrum
Laplacian spectrum
Algebraic connectivity
Binomial matrix
Spectral radius
Walk-regular graph
topic Token graph
Adjacency spectrum
Local spectrum
Laplacian spectrum
Algebraic connectivity
Binomial matrix
Spectral radius
Walk-regular graph
description Let G be a graph on n vertices. The k-token graph (or symmetric k-th power) of G, denoted by Fk(G), has as vertices the (n/k) k-subsets of vertices from G, and two vertices are adjacent when their symmetric difference is a pair of adjacent vertices in G. In particular, Fk(Kn) is the Johnson graph J(n, k), which is a distance-regular graph used in coding theory. In this paper, we present some results concerning the (adjacency and Laplacian) spectrum of Fk(G) in terms of the spectrum of G. For instance, when G is walk-regular, an exact value for the spectral radius (or maximum eigenvalue) of Fk(G) is obtained. When G is distance-regular, other eigenvalues of its 2-token graph are derived using the theory of equitable partitions. A generalization of Aldous’ spectral gap conjecture (which is now a theorem) is proposed.
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/10.1007/s40590-023-00583-3
https://hdl.handle.net/10459.1/464851
url https://doi.org/10.1007/s40590-023-00583-3
https://hdl.handle.net/10459.1/464851
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/PGC2018-095471-B-I00
Reproducció del document publicat a http://doi.org/10.1007/s40590-023-00583-3
Boletín de la Sociedad Matemática Mexicana, 2024, vol. 30, art. 11
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 Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:Repositori Obert UdL
instname:Universitat de Lleida (UdL)
instname_str Universitat de Lleida (UdL)
reponame_str Repositori Obert UdL
collection Repositori Obert UdL
repository.name.fl_str_mv
repository.mail.fl_str_mv
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