Non existence of almost Moore digraphs of diameter three

Almost 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....

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Authors: Conde Colom, Josep, Gimbert i Quintilla, Joan, Miret, Josep M. (Josep Maria), Moreno Chiral, Ramiro, Gonzàlez, J.
Format: article
Status:Published version
Publication Date:2008
Country:España
Institution:Universitat de Lleida (UdL)
Repository:Repositori Obert UdL
OAI Identifier:oai:repositori.udl.cat:10459.1/49273
Online Access:http://hdl.handle.net/10459.1/49273
Access Level:Open access
Keyword:Almost Moore digraph
Characteristic polynomial
Cyclotomic polynomial
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spelling Non existence of almost Moore digraphs of diameter threeConde Colom, JosepGimbert i Quintilla, JoanMiret, Josep M. (Josep Maria)Moreno Chiral, RamiroGonzàlez, J.Almost Moore digraphCharacteristic polynomialCyclotomic polynomialAlmost 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.Partially supported by the Ministry of Science and Technology, Spain, under the projects TIC2003- 09188, MTM2006-15038-C02-02 and MTM2007-66842-C02-02.Electronic Journal of Combinatorics2008info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://hdl.handle.net/10459.1/49273reponame:Repositori Obert UdL instname:Universitat de Lleida (UdL)Inglésinfo:eu-repo/grantAgreement/MICYT//TIC2003-09188info:eu-repo/grantAgreement/MEC//MTM2006-15038-C02-02info:eu-repo/grantAgreement/MEC//MTM2007-66842-C02-02Reproducció del document publicat a http://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r87Electronic Journal of Combinatorics, 2008, vol. 15, núm. 1(c) Conde et al., 2008info:eu-repo/semantics/openAccessoai:repositori.udl.cat:10459.1/492732026-06-24T12:42:17Z
dc.title.none.fl_str_mv Non existence of almost Moore digraphs of diameter three
title Non existence of almost Moore digraphs of diameter three
spellingShingle Non existence of almost Moore digraphs of diameter three
Conde Colom, Josep
Almost Moore digraph
Characteristic polynomial
Cyclotomic polynomial
title_short Non existence of almost Moore digraphs of diameter three
title_full Non existence of almost Moore digraphs of diameter three
title_fullStr Non existence of almost Moore digraphs of diameter three
title_full_unstemmed Non existence of almost Moore digraphs of diameter three
title_sort Non existence of almost Moore digraphs of diameter three
dc.creator.none.fl_str_mv Conde Colom, Josep
Gimbert i Quintilla, Joan
Miret, Josep M. (Josep Maria)
Moreno Chiral, Ramiro
Gonzàlez, J.
author Conde Colom, Josep
author_facet Conde Colom, Josep
Gimbert i Quintilla, Joan
Miret, Josep M. (Josep Maria)
Moreno Chiral, Ramiro
Gonzàlez, J.
author_role author
author2 Gimbert i Quintilla, Joan
Miret, Josep M. (Josep Maria)
Moreno Chiral, Ramiro
Gonzàlez, J.
author2_role author
author
author
author
dc.subject.none.fl_str_mv Almost Moore digraph
Characteristic polynomial
Cyclotomic polynomial
topic Almost Moore digraph
Characteristic polynomial
Cyclotomic polynomial
description Almost 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.
publishDate 2008
dc.date.none.fl_str_mv 2008
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 http://hdl.handle.net/10459.1/49273
url http://hdl.handle.net/10459.1/49273
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv info:eu-repo/grantAgreement/MICYT//TIC2003-09188
info:eu-repo/grantAgreement/MEC//MTM2006-15038-C02-02
info:eu-repo/grantAgreement/MEC//MTM2007-66842-C02-02
Reproducció del document publicat a http://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r87
Electronic Journal of Combinatorics, 2008, vol. 15, núm. 1
dc.rights.none.fl_str_mv (c) Conde et al., 2008
info:eu-repo/semantics/openAccess
rights_invalid_str_mv (c) Conde et al., 2008
eu_rights_str_mv openAccess
dc.publisher.none.fl_str_mv Electronic Journal of Combinatorics
publisher.none.fl_str_mv Electronic Journal of Combinatorics
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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