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dc.contributor.authorConde Colom, Josep
dc.contributor.authorGimbert i Quintilla, Joan
dc.contributor.authorMiret, Josep M. (Josep Maria)
dc.contributor.authorMoreno Chiral, Ramiro
dc.contributor.authorGonzàlez, J.
dc.date.accessioned2015-12-18T13:19:17Z
dc.date.available2015-12-18T13:19:17Z
dc.date.issued2008
dc.identifier.issn1077-8926
dc.identifier.urihttp://hdl.handle.net/10459.1/49273
dc.description.abstractAlmost Moore digraphs appear in the context of the degree/diameter problem as a class of extremal directed graphs, in the sense that their order is one less than the unattainable Moore bound M(d; k) = 1 + d + + dk, where d > 1 and k > 1 denote the maximum out-degree and diameter, respectively. So far, the problem of their existence has only been solved when d = 2; 3 or k = 2. In this paper, we prove that almost Moore digraphs of diameter k = 3 do not exist for any degree d. The enumeration of almost Moore digraphs of degree d and diameter k = 3 turns out to be equivalent to the search of binary matrices A ful lling that AJ = dJ and I+A+A2+A3 = J +P, where J denotes the all-one matrix and P is a permutation matrix . We use spectral techniques in order to show that such equation has no (0; 1)-matrix solutions. More precisely, we obtain the factorization in Q[x] of the characteristic polynomial of A, in terms of the cycle structure of P, we compute the trace of A and we derive a contradiction on some algebraic multiplicities of the eigenvalues of A. In order to get the factorization of det(xI - A) we determine when the polynomials Fn(x) = n(1 + x + x2 + x3) are irreducible in Q[x], where n(x) denotes the n-th cyclotomic polynomial, since in such case they become `big pieces' of det(xI - A). By using concepts and techniques from algebraic number theory, we prove that Fn(x) is always irreducible in Q[x], unless n = 1; 10. So, by combining tools from matrix and number theory we have been able to solve a problem of graph theory.ca_ES
dc.description.sponsorshipPartially supported by the Ministry of Science and Technology, Spain, under the projects TIC2003- 09188, MTM2006-15038-C02-02 and MTM2007-66842-C02-02.
dc.language.isoengca_ES
dc.publisherElectronic Journal of Combinatoricsca_ES
dc.relationMICYT/PN2000-2003/TIC2003-09188
dc.relationMIECI/PN2004-2007/MTM2006-15038-C02-02
dc.relationMIECI/PN2004-2007/MTM2007-66842-C02-02
dc.relation.isformatofReproducció del document publicat a http://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r87ca_ES
dc.relation.ispartofElectronic Journal of Combinatorics, 2008, vol. 15, núm. 1ca_ES
dc.rights(c) Conde et al., 2008ca_ES
dc.subjectAlmost Moore digraphca_ES
dc.subjectCharacteristic polynomialca_ES
dc.subjectCyclotomic polynomialca_ES
dc.titleNon existence of almost Moore digraphs of diameter threeca_ES
dc.typearticleca_ES
dc.identifier.idgrec012456
dc.type.versionpublishedVersionca_ES
dc.rights.accessRightsinfo:eu-repo/semantics/openAccessca_ES


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