Counting independent sets in cubic graphs of given girth

We prove a tight upper bound on the independence polynomial (and total number of independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the Heawood graph, the point/line incidence graph of the Fano plane. We also give a tight lower bound on the total number of inde...

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Detalles Bibliográficos
Autores: Perarnau Llobet, Guillem|||0000-0002-1953-9511, Perkins, Will
Tipo de recurso: artículo
Fecha de publicación:2018
País:España
Institución:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/133725
Acceso en línea:https://hdl.handle.net/2117/133725
https://dx.doi.org/10.1016/j.jctb.2018.04.009
Access Level:acceso abierto
Palabra clave:Graph theory
independent sets
independence polynomial
hard-core model
Petersen graph
Heawood graph
occupancy fraction
Grafs, Teoria de
Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs
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spelling Counting independent sets in cubic graphs of given girthPerarnau Llobet, Guillem|||0000-0002-1953-9511Perkins, WillGraph theoryindependent setsindependence polynomialhard-core modelPetersen graphHeawood graphoccupancy fractionGrafs, Teoria deÀrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafsWe prove a tight upper bound on the independence polynomial (and total number of independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the Heawood graph, the point/line incidence graph of the Fano plane. We also give a tight lower bound on the total number of independent sets of triangle-free cubic graphs. This bound is achieved by unions of the Petersen graph. We conjecture that in fact all Moore graphs are extremal for the scaled number of independent sets in regular graphs of a given minimum girth, maximizing this quantity if their girth is even and minimizing if odd. The Heawood and Petersen graphs are instances of this conjecture, along with complete graphs, complete bipartite graphs, and cycles.20182018-09-2620192019-05-30journal articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/133725https://dx.doi.org/10.1016/j.jctb.2018.04.009reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial-NoDerivs 3.0 Spainhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/1337252026-05-27T15:37:01Z
dc.title.none.fl_str_mv Counting independent sets in cubic graphs of given girth
title Counting independent sets in cubic graphs of given girth
spellingShingle Counting independent sets in cubic graphs of given girth
Perarnau Llobet, Guillem|||0000-0002-1953-9511
Graph theory
independent sets
independence polynomial
hard-core model
Petersen graph
Heawood graph
occupancy fraction
Grafs, Teoria de
Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs
title_short Counting independent sets in cubic graphs of given girth
title_full Counting independent sets in cubic graphs of given girth
title_fullStr Counting independent sets in cubic graphs of given girth
title_full_unstemmed Counting independent sets in cubic graphs of given girth
title_sort Counting independent sets in cubic graphs of given girth
dc.creator.none.fl_str_mv Perarnau Llobet, Guillem|||0000-0002-1953-9511
Perkins, Will
author Perarnau Llobet, Guillem|||0000-0002-1953-9511
author_facet Perarnau Llobet, Guillem|||0000-0002-1953-9511
Perkins, Will
author_role author
author2 Perkins, Will
author2_role author
dc.subject.none.fl_str_mv Graph theory
independent sets
independence polynomial
hard-core model
Petersen graph
Heawood graph
occupancy fraction
Grafs, Teoria de
Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs
topic Graph theory
independent sets
independence polynomial
hard-core model
Petersen graph
Heawood graph
occupancy fraction
Grafs, Teoria de
Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs
description We prove a tight upper bound on the independence polynomial (and total number of independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the Heawood graph, the point/line incidence graph of the Fano plane. We also give a tight lower bound on the total number of independent sets of triangle-free cubic graphs. This bound is achieved by unions of the Petersen graph. We conjecture that in fact all Moore graphs are extremal for the scaled number of independent sets in regular graphs of a given minimum girth, maximizing this quantity if their girth is even and minimizing if odd. The Heawood and Petersen graphs are instances of this conjecture, along with complete graphs, complete bipartite graphs, and cycles.
publishDate 2018
dc.date.none.fl_str_mv 2018
2018-09-26
2019
2019-05-30
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/133725
https://dx.doi.org/10.1016/j.jctb.2018.04.009
url https://hdl.handle.net/2117/133725
https://dx.doi.org/10.1016/j.jctb.2018.04.009
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.300719

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