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A new class of polynomials from the spectrum of a graph, and its application to bound the k-independence number
dc.contributor.author | Fiol Mora, Miquel Àngel |
dc.contributor.other | Universitat Politècnica de Catalunya. Departament de Matemàtiques |
dc.date.accessioned | 2020-12-17T09:45:59Z |
dc.date.available | 2022-11-15T01:33:11Z |
dc.date.issued | 2020-11-15 |
dc.identifier.citation | Fiol, M. A new class of polynomials from the spectrum of a graph, and its application to bound the k-independence number. "Linear algebra and its applications", 15 Novembre 2020, vol. 605, p. 1-20. |
dc.identifier.issn | 0024-3795 |
dc.identifier.other | https://arxiv.org/pdf/1907.08626.pdf |
dc.identifier.uri | http://hdl.handle.net/2117/334568 |
dc.description | © 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ |
dc.description.abstract | The k-independence number of a graph is the maximum size of a set of vertices at pairwise distance greater than k. A graph is called k-partially walk-regular if the number of closed walks of a given length l = k, rooted at a vertex v, only depends on l. In particular, a distance-regular graph is also k-partially walk-regular for any k. In this paper, we introduce a new family of polynomials obtained from the spectrum of a graph, called minor polynomials. These polynomials, together with the interlacing technique, allow us to give tight spectral bounds on the k-independence number of a k-partially walk-regular graph. With some examples and infinite families of graphs whose bounds are tight, we also show that the odd graph O` with ` odd has no 1-perfect code. Moreover, we show that our bound coincide, in general, with the Delsarte’s linear programming bound and the Lovász theta number ¿, the best ones to our knowledge. In fact, as a byproduct, it is shown that the given bounds also apply for ¿ and T, the Shannon capacity of a graph. Moreover, apart from the possible interest that the minor polynomials can have, our approach has the advantage of yielding closed formulas and, so, allowing asymptotic analysis. |
dc.description.sponsorship | This research 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. |
dc.format.extent | 20 p. |
dc.language.iso | eng |
dc.publisher | Elsevier |
dc.subject | Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Combinatòria |
dc.subject.lcsh | Graph theory |
dc.subject.other | Graph |
dc.subject.other | k-Independence number |
dc.subject.other | Spectrum |
dc.subject.other | Interlacing |
dc.subject.other | Minor polynomial |
dc.subject.other | k-Partially walk-regular |
dc.subject.other | Delsarte's LP bound |
dc.subject.other | Lovász theta number |
dc.subject.other | Shannon capacity |
dc.title | A new class of polynomials from the spectrum of a graph, and its application to bound the k-independence number |
dc.type | Article |
dc.subject.lemac | Grafs, Teoria de |
dc.contributor.group | Universitat Politècnica de Catalunya. COMBGRAPH - Combinatòria, Teoria de Grafs i Aplicacions |
dc.identifier.doi | 10.1016/j.laa.2020.07.009 |
dc.description.peerreviewed | Peer Reviewed |
dc.subject.ams | Classificació AMS::05 Combinatorics::05C Graph theory |
dc.relation.publisherversion | https://www.sciencedirect.com/science/article/abs/pii/S0024379520303323?via%3Dihub |
dc.rights.access | Open Access |
local.identifier.drac | 28924818 |
dc.description.version | Postprint (author's final draft) |
dc.relation.projectid | info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PGC2018-095471-B-I00/ES/ESTUDIO MATEMATICO DE LOS FALLOS EN CASCADA EN SISTEMAS COMPLEJOS MEDIANTE INVARIANTES Y CENTRALIDADES EN GRAFOS. APLICACIONES A REDES REALES./ |
local.citation.author | Fiol, M. |
local.citation.publicationName | Linear algebra and its applications |
local.citation.volume | 605 |
local.citation.startingPage | 1 |
local.citation.endingPage | 20 |
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