Nonexistence of almost Moore digraphs of diameter four

Regular digraphs of degree d > 1, diameter k > 1 and order N(d; k) = d+ +dk will be called almost Moore (d; k)-digraphs. So far, the problem of their existence has only been solved when d = 2; 3 or k = 2; 3. In this paper we prove that almost Moore digraphs of diameter 4 do not exist for any degree dPostprint (published version

Contribucions a l'estudi dels grafs i digrafs propers als de Moore

El principal objectiu d'aquesta tesi és el de contribuir a l'estudi de l'existència i classificació dels grafs i digrafs que puguin admetre el màxim nombre de vèrtexs sota determinades condicions donats el grau i el diàmetre. Aquest estudi consta de tres parts ben diferenciades, una sobre digrafs i dos sobre grafs. En el treball relacionat amb els digrafs demostrem que els digrafs quasi de Moore de diàmetre k = 3 i qualsevol grau no existeixen. Així mateix provem la no existència dels digrafs quasi de Moore de diàmetre 4 i qualsevol grau assumint la irreductibilitat en Q[x] de certs polinomis. En quan als grafs ens hem centrat en l'existència dels de grau d, diàmetre 2 i defecte 2, anomenats (d,2,2)-grafs i assumint la irreductibilitat en Q[x] de certs polinomis provem que no existeixen per a cap grau. A més provem que no existeixen per a graus entre 4 i 50. Finalment estudiem els grafs radials de Moore de grau d i radi k. Proposem diferents mesures per classificar-los d'acord a la proximitat de les seves propietats a les d'un graf de Moore i ordenem segons aquestes mesures tots els grafs radials de Moore en els casos (d,k) = {(3,2), (3,3), (4,2)}.El principal objetivo de esta tesis es el de contribuir al estudio de la existencia y clasificación de los grafos y digrafos que puedan admitir el máximo número de vértices bajo determinadas condiciones dados el grado y el diámetro. Este estudio consta de tres partes bien diferenciadas, una sobre digrafos y dos sobre grafos. En el trabajo relacionado con los digrafos demostramos que los digrafos casi de Moore de diámetro k = 3 y cualquier grado no existen. Asimismo probamos la no existencia de los digrafos casi de Moore de diámetro 4 y cualquier grado suponiendo la irreducibilidad en Q[x] de ciertos polinomios. En cuanto a los grafos nos hemos centrado en la existencia de los de grado d, diámetro 2 y defecto 2, llamados (d,2,2)-grafos y suponiendo la irreducibilidad en Q[x] de ciertos polinomios probamos que no existen para ningún grado. Además probamos que no existen para grados entre 4 y 50. Finalmente estudiamos los grafos radiales de Moore de grado d y radio k. Proponemos diferentes medidas para clasificarlos de acuerdo a la proximidad de sus propiedades a las de un grafo de Moore y ordenamos según estas medidas todos los grafos radiales de Moore en los casos (d, k) = {(3,2), (3,3), (4,2)}.The main goal of this thesis is to contribute to the study of the existence and classification of graphs and digraphs that can achieve the maximum number of vertices under certain conditions given the degree and the diameter. This study consists of three differenciated parts, one on digraphs and two on graphs. The work on digraphs focuses on almost Moore digraphs. We prove that they do not exist for diameter 3 and any degree. Besides, we prove the non-existence of almost Moore digraphs of diameter 4 assuming the irreducibility in Q[x] of certain polynomials. Concerning graphs, we discuss the existence of graphs of degree d, diameter 2 and defect 2. Assuming the irreducibility in Q[x] of certain polynomials we prove their non existence. We also show they do not exist for degrees between 4 and 50. Finally we study radial Moore graphs of degree d and radius k. We propose different measures for classifying them in terms of their proximity to extremal properties of a Moore graph. By means of our measures, we are able to enumerate all radial Moore graphs for the cases (d, k) = {(3.2), (3.3), (4.2)}

On the nonexistence of almost Moore digraphs

Digraphs of maximum out-degree at most d > 1, diameter at most k > 1 and order N(d, k) = d + ... + d(k) are called almost Moore or (d, k)-digraphs. So far, the problem of their existence has been solved only when d = 2, 3 or k = 2, 3, 4. In this paper we derive the nonexistence of (d, k)-digraphs, with k > 4 and d > 3, under the assumption of a conjecture related to the factorization of the polynomials Phi(n)(1 + x + ... + x(k)), where Phi(n)(x) denotes the nth cyclotomic polynomial and 1 < n <= N(d, k). Moreover, we prove that almost Moore digraphs do not exist for the particular cases when k = 5 and d = 4, 5 or 6. (C) 2014 Elsevier Ltd. All rights reserved.Postprint (published version

On the nonexistence of almost Moore digraphs

Digraphs of maximum out-degree at most d > 1, diameter at most k > 1 and order N(d, k) = d + ... + d(k) are called almost Moore or (d, k)-digraphs. So far, the problem of their existence has been solved only when d = 2, 3 or k = 2, 3, 4. In this paper we derive the nonexistence of (d, k)-digraphs, with k > 4 and d > 3, under the assumption of a conjecture related to the factorization of the polynomials Phi(n)(1 + x + ... + x(k)), where Phi(n)(x) denotes the nth cyclotomic polynomial and 1 < n <= N(d, k). Moreover, we prove that almost Moore digraphs do not exist for the particular cases when k = 5 and d = 4, 5 or 6. (C) 2014 Elsevier Ltd. All rights reserved

Nonexistence of almost Moore digraphs of diameter four

Regular digraphs of degree d > 1, diameter k > 1 and order N(d; k) = d+ +dk will be called almost Moore (d; k)-digraphs. So far, the problem of their existence has only been solved when d = 2; 3 or k = 2; 3. In this paper we prove that almost Moore digraphs of diameter 4 do not exist for any degree

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. 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

State-based predictions with self-correction on Enterprise Desktop Grid environments

The abundant computing resources in current organizations provide new opportunities for executing parallel scientific applications and using resources. The Enterprise Desktop Grid Computing (EDGC) paradigm addresses the potential for harvesting the idle computing resources of an organization’s desktop PCs to support the execution of the company’s large-scale applications. In these environments, the accuracy of response-time predictions is essential for effective metascheduling that maximizes resource usage without harming the performance of the parallel and local applications. However, this accuracy is a major challenge due to the heterogeneity and non-dedicated nature of EDGC resources. In this paper, two new prediction techniques are presented based on the state of resources. A thorough analysis by linear regression demonstrated that the proposed techniques capture the real behavior of the parallel applications better than other common techniques in the literature. Moreover, it is possible to reduce deviations with a proper modeling of prediction errors, and thus, a Self-adjustable Correction method (SAC) for detecting and correcting the prediction deviations was proposed with the ability to adapt to the changes in load conditions. An extensive evaluation in a real environment was conducted to validate the SAC method. The results show that the use of SAC increases the accuracy of response-time predictions by 35%. The cost of predictions with self-correction and its accuracy in a real environment was analyzed using a combination of the proposed techniques. The results demonstrate that the cost of predictions is negligible and the combined use of the prediction techniques is preferable