3er. 2010
http://hdl.handle.net/2099/10267
Facultat de Matemàtiques i Estadística, Universitat Politècnica de Catalunya, Barcelona, 9-11 June 2010Mon, 30 Nov 2015 17:37:08 GMT2015-11-30T17:37:08ZTopology of Cayley graphs applied to inverse additive problems
http://hdl.handle.net/2099/10390
Topology of Cayley graphs applied to inverse additive problems
Hamidoune, Yahya Ould
We present the basic isopermetric structure theory, obtaining some new simplified proofs. Let 1 ≤ r ≤ k be integers. As an
application, we obtain simple descriptions for the subsets S of an abelian group with |kS| ≤ k|S|−k+1 or |kS−rS|−(k+r)|S|, where where S denotes as usual the Minkowski sum of copies of S. These results may be applied to several questions in Combinatorics and Additive Combinatorics including the Frobenius Problem, Waring’s problem in finite fields and the structure of abelian Cayley graphs with a big diameter.
Thu, 12 May 2011 12:31:04 GMThttp://hdl.handle.net/2099/103902011-05-12T12:31:04ZHamidoune, Yahya OuldWe present the basic isopermetric structure theory, obtaining some new simplified proofs. Let 1 ≤ r ≤ k be integers. As an
application, we obtain simple descriptions for the subsets S of an abelian group with |kS| ≤ k|S|−k+1 or |kS−rS|−(k+r)|S|, where where S denotes as usual the Minkowski sum of copies of S. These results may be applied to several questions in Combinatorics and Additive Combinatorics including the Frobenius Problem, Waring’s problem in finite fields and the structure of abelian Cayley graphs with a big diameter.Algebraic characterizations of bipartite distance-regular graphs
http://hdl.handle.net/2099/10389
Algebraic characterizations of bipartite distance-regular graphs
Fiol Mora, Miquel Àngel
Bipartite graphs are combinatorial objects bearing some interesting symmetries. Thus, their spectra—eigenvalues of its adjacency
matrix—are symmetric about zero, as the corresponding eigenvectors come into pairs. Moreover, vertices in the same (respectively, different) independent set are always at even (respectively, odd) distance. Both properties have well-known consequences in most properties and parameters of such graphs.
Roughly speaking, we could say that the conditions for a given property to hold in a general graph can be somehow relaxed to guaranty the same property for a bipartite graph. In this paper we comment upon this phenomenon in the framework of distance-regular graphs for which several characterizations, both of combinatorial or algebraic nature, are known. Thus, the
presented characterizations of bipartite distance-regular graphs involve such parameters as the numbers of walks between vertices (entries of the powers of the adjacency matrix A), the crossed local multiplicities (entries of the idempotents $E_i$ or eigenprojectors), the predistance polynomials, etc. For instance, it is known that a graph G, with eigenvalues $λ_0$ > $λ_1$ > · · · > $λ_d$ and diameter D = d, is distance-regular if and only if its idempotents $E_1$ and $E_d$ belong to the vector space D spanned by its distance matrices I,A,$A_2$, . . .$A_d$. In contrast with this, for the same result to be true in the case of bipartite graphs, only $E_1$ ∈ D need to be required.
Thu, 12 May 2011 12:21:29 GMThttp://hdl.handle.net/2099/103892011-05-12T12:21:29ZFiol Mora, Miquel ÀngelBipartite graphs are combinatorial objects bearing some interesting symmetries. Thus, their spectra—eigenvalues of its adjacency
matrix—are symmetric about zero, as the corresponding eigenvectors come into pairs. Moreover, vertices in the same (respectively, different) independent set are always at even (respectively, odd) distance. Both properties have well-known consequences in most properties and parameters of such graphs.
Roughly speaking, we could say that the conditions for a given property to hold in a general graph can be somehow relaxed to guaranty the same property for a bipartite graph. In this paper we comment upon this phenomenon in the framework of distance-regular graphs for which several characterizations, both of combinatorial or algebraic nature, are known. Thus, the
presented characterizations of bipartite distance-regular graphs involve such parameters as the numbers of walks between vertices (entries of the powers of the adjacency matrix A), the crossed local multiplicities (entries of the idempotents $E_i$ or eigenprojectors), the predistance polynomials, etc. For instance, it is known that a graph G, with eigenvalues $λ_0$ > $λ_1$ > · · · > $λ_d$ and diameter D = d, is distance-regular if and only if its idempotents $E_1$ and $E_d$ belong to the vector space D spanned by its distance matrices I,A,$A_2$, . . .$A_d$. In contrast with this, for the same result to be true in the case of bipartite graphs, only $E_1$ ∈ D need to be required.A mathematical model for dynamic memory networks
http://hdl.handle.net/2099/10388
A mathematical model for dynamic memory networks
Fàbrega Canudas, José; Fiol Mora, Miquel Àngel; Serra Albó, Oriol; Andrés Yebra, José Luis
The aim of this paper is to bring together the work done several years ago by M.A. Fiol and the other authors to formulate a quite general mathematical model for a kind of permutation networks known as dynamic memories. A dynamic memory is constituted by an array of cells, each storing one datum, and an interconnection network between the cells that allows the constant circulation of the stored data. The objective is to
design the interconnection network in order to have short access time and a simple memory control. We review how most of the proposals of dynamic memories that have appeared in the literature fit in this general model, and how it can be used to
design new structures with good access properties. Moreover, using the idea of projecting a digraph onto a de Bruijn digraph, we propose new structures for dynamic memories with vectorial capabilities. Some of these new proposals are based on iterated line digraphs, which have been widely and successfully used by
M.A. Fiol and his coauthors to solve many different problems in graph theory.
Thu, 12 May 2011 12:09:12 GMThttp://hdl.handle.net/2099/103882011-05-12T12:09:12ZFàbrega Canudas, JoséFiol Mora, Miquel ÀngelSerra Albó, OriolAndrés Yebra, José LuisThe aim of this paper is to bring together the work done several years ago by M.A. Fiol and the other authors to formulate a quite general mathematical model for a kind of permutation networks known as dynamic memories. A dynamic memory is constituted by an array of cells, each storing one datum, and an interconnection network between the cells that allows the constant circulation of the stored data. The objective is to
design the interconnection network in order to have short access time and a simple memory control. We review how most of the proposals of dynamic memories that have appeared in the literature fit in this general model, and how it can be used to
design new structures with good access properties. Moreover, using the idea of projecting a digraph onto a de Bruijn digraph, we propose new structures for dynamic memories with vectorial capabilities. Some of these new proposals are based on iterated line digraphs, which have been widely and successfully used by
M.A. Fiol and his coauthors to solve many different problems in graph theory.Large Edge-non-vulnerable Graphs
http://hdl.handle.net/2099/10387
Large Edge-non-vulnerable Graphs
Delorme, Charles
In this paper we study the graphs such that the deletion of any edge does not increase the diameter. We give some upper bounds for the order of such a graph with given maximum degree and diameter. On the other hand construction of graphs provide lower bounds. As usual, for this kind of problems, there
is often a gap between these two bounds.
Thu, 12 May 2011 11:55:28 GMThttp://hdl.handle.net/2099/103872011-05-12T11:55:28ZDelorme, CharlesIn this paper we study the graphs such that the deletion of any edge does not increase the diameter. We give some upper bounds for the order of such a graph with given maximum degree and diameter. On the other hand construction of graphs provide lower bounds. As usual, for this kind of problems, there
is often a gap between these two bounds.Dual concepts of almost distance-regularity and the spectral excess theorem
http://hdl.handle.net/2099/10386
Dual concepts of almost distance-regularity and the spectral excess theorem
Dalfó Simó, Cristina; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest; Dam, Edwin R. van
Generally speaking, ‘almost distance-regular’ graphs are graphs that share some, but not necessarily all, regularity properties that characterize distance-regular graphs. In this paper we first
propose two dual concepts of almost distance-regularity. In some cases, they coincide with concepts introduced before by other authors, such as partially distance-regular graphs. Our study focuses on finding out when almost distance-regularity leads to distance-regularity. In particular, some ‘economic’ (in the sense of minimizing the number of conditions) old and new
characterizations of distance-regularity are discussed. Moreover, other characterizations based on the average intersection numbers and the recurrence coefficients are obtained. In some cases, our results can also be seen as a generalization of the
so-called spectral excess theorem for distance-regular graphs.
Thu, 12 May 2011 11:06:15 GMThttp://hdl.handle.net/2099/103862011-05-12T11:06:15ZDalfó Simó, CristinaFiol Mora, Miquel ÀngelGarriga Valle, ErnestDam, Edwin R. vanGenerally speaking, ‘almost distance-regular’ graphs are graphs that share some, but not necessarily all, regularity properties that characterize distance-regular graphs. In this paper we first
propose two dual concepts of almost distance-regularity. In some cases, they coincide with concepts introduced before by other authors, such as partially distance-regular graphs. Our study focuses on finding out when almost distance-regularity leads to distance-regularity. In particular, some ‘economic’ (in the sense of minimizing the number of conditions) old and new
characterizations of distance-regularity are discussed. Moreover, other characterizations based on the average intersection numbers and the recurrence coefficients are obtained. In some cases, our results can also be seen as a generalization of the
so-called spectral excess theorem for distance-regular graphs.Large digraphs of given diameter and degree from coverings
http://hdl.handle.net/2099/10385
Large digraphs of given diameter and degree from coverings
Ždímalová, Mária; Staneková, Lubica
We show that a construction of Comellas and Fiol for large vertex-transitive digraphs of given degree and diameter from small digraphs preserves the properties of being a Cayley digraph and being a regular covering.
Thu, 12 May 2011 10:52:54 GMThttp://hdl.handle.net/2099/103852011-05-12T10:52:54ZŽdímalová, MáriaStaneková, LubicaWe show that a construction of Comellas and Fiol for large vertex-transitive digraphs of given degree and diameter from small digraphs preserves the properties of being a Cayley digraph and being a regular covering.Fiedler’s Clustering on m-dimensional Lattice Graphs
http://hdl.handle.net/2099/10384
Fiedler’s Clustering on m-dimensional Lattice Graphs
Trajanovski, Stojan; Mieghem, Piet Van
We consider the partitioning of m-dimensional lattice graphs using Fiedler’s approach [1], that requires the determination of the eigenvector belonging to the second smallest eigenvalue of
the Laplacian. We examine the general m-dimensional lattice and, in particular, the special cases: the 1-dimensional path graph $P_N$ and the 2-dimensional lattice graph. We determine the size of the clusters and the number of links, which are cut by this partitioning as a function of Fiedler’s threshold α.
Thu, 12 May 2011 10:46:55 GMThttp://hdl.handle.net/2099/103842011-05-12T10:46:55ZTrajanovski, StojanMieghem, Piet VanWe consider the partitioning of m-dimensional lattice graphs using Fiedler’s approach [1], that requires the determination of the eigenvector belonging to the second smallest eigenvalue of
the Laplacian. We examine the general m-dimensional lattice and, in particular, the special cases: the 1-dimensional path graph $P_N$ and the 2-dimensional lattice graph. We determine the size of the clusters and the number of links, which are cut by this partitioning as a function of Fiedler’s threshold α.Large graphs of diameter two and given degree
http://hdl.handle.net/2099/10383
Large graphs of diameter two and given degree
Širáň, Jozef; Siagiová, Jana; Ždímalová, Mária
Let r(d, 2), C(d, 2), and AC(d, 2) be the largest order of a regular graph, a Cayley graph, and a Cayley graph of an Abelian
group, respectively, of diameter 2 and degree d. The best currently known lower bounds on these parameters are r(d, 2) ≥
$d^2$ − d + 1 for d − 1 an odd prime power (with a similar result for powers of two), C(d, 2) ≥ (d + 1)$^2$/2 for degrees d = 2q − 1
where q is an odd prime power, and AC(d, 2) ≥ (3/8)($d^2$ − 4) where d = 4q − 2 for an odd prime power q.
Using a number theory result on distribution of primes we prove, for all sufficiently large d, lower bounds on r(d, 2), C(d, 2), and AC(d, 2) of the form c · $d^2$ − O($d^1.525$) for c = 1, 1/2, and 3/8,
respectively. We also prove results of a similar flavour for vertex transitive
graphs and Cayley graphs of cyclic groups.
Thu, 12 May 2011 10:36:44 GMThttp://hdl.handle.net/2099/103832011-05-12T10:36:44ZŠiráň, JozefSiagiová, JanaŽdímalová, MáriaLet r(d, 2), C(d, 2), and AC(d, 2) be the largest order of a regular graph, a Cayley graph, and a Cayley graph of an Abelian
group, respectively, of diameter 2 and degree d. The best currently known lower bounds on these parameters are r(d, 2) ≥
$d^2$ − d + 1 for d − 1 an odd prime power (with a similar result for powers of two), C(d, 2) ≥ (d + 1)$^2$/2 for degrees d = 2q − 1
where q is an odd prime power, and AC(d, 2) ≥ (3/8)($d^2$ − 4) where d = 4q − 2 for an odd prime power q.
Using a number theory result on distribution of primes we prove, for all sufficiently large d, lower bounds on r(d, 2), C(d, 2), and AC(d, 2) of the form c · $d^2$ − O($d^1.525$) for c = 1, 1/2, and 3/8,
respectively. We also prove results of a similar flavour for vertex transitive
graphs and Cayley graphs of cyclic groups.An overview of the degree/diameter problem for directed, undirected and mixed graphs
http://hdl.handle.net/2099/10382
An overview of the degree/diameter problem for directed, undirected and mixed graphs
Miller, Mirka
A well-known fundamental problem in extremal graph theory is the degree/diameter problem, which is to determine the largest (in terms of the number of vertices) graphs or digraphs or mixed graphs of given maximum degree, respectively, maximum outdegree,
respectively, mixed degree; and given diameter. General upper bounds, called Moore bounds, exist for the largest possible
order of such graphs, digraphs and mixed graphs of given maximum degree d (respectively, maximum out-degree d, respectively, maximum mixed degree) and diameter k.
In recent years, there have been many interesting new results in all these three versions of the problem, resulting in improvements in both the lower bounds and the upper bounds on the largest possible number of vertices. However, quite a number of questions regarding the degree/diameter problem are still wide open. In this paper we present an overview of the current state
of the degree/diameter problem, for undirected, directed and mixed graphs, and we outline several related open problems.
Thu, 12 May 2011 10:23:58 GMThttp://hdl.handle.net/2099/103822011-05-12T10:23:58ZMiller, MirkaA well-known fundamental problem in extremal graph theory is the degree/diameter problem, which is to determine the largest (in terms of the number of vertices) graphs or digraphs or mixed graphs of given maximum degree, respectively, maximum outdegree,
respectively, mixed degree; and given diameter. General upper bounds, called Moore bounds, exist for the largest possible
order of such graphs, digraphs and mixed graphs of given maximum degree d (respectively, maximum out-degree d, respectively, maximum mixed degree) and diameter k.
In recent years, there have been many interesting new results in all these three versions of the problem, resulting in improvements in both the lower bounds and the upper bounds on the largest possible number of vertices. However, quite a number of questions regarding the degree/diameter problem are still wide open. In this paper we present an overview of the current state
of the degree/diameter problem, for undirected, directed and mixed graphs, and we outline several related open problems.On the k-restricted edge-connectivity of matched sum graphs
http://hdl.handle.net/2099/10381
On the k-restricted edge-connectivity of matched sum graphs
Marcote Ordax, Francisco Javier
A matched sum graph $G_1$M$G_2$ of two graphs $G_1$ and $G_2$ of the same order n is obtained by adding to the union (or sum) of $G_1$ and $G_2$ a set M of n independent edges which join vertices in V ($G_1$) to vertices in V ($G_2$). When $G_1$ and $G_2$ are isomorphic, $G_1$M$G_2$ is just a permutation graph. In this work we derive
bounds for the k-restricted edge connectivity λ(k) of matched sum graphs $G_1$M$G_2$ for 2 ≤ k ≤ 5, and present some sufficient conditions for the optimality of λ(k)($G_1$M$G_2$).
Thu, 12 May 2011 10:11:29 GMThttp://hdl.handle.net/2099/103812011-05-12T10:11:29ZMarcote Ordax, Francisco JavierA matched sum graph $G_1$M$G_2$ of two graphs $G_1$ and $G_2$ of the same order n is obtained by adding to the union (or sum) of $G_1$ and $G_2$ a set M of n independent edges which join vertices in V ($G_1$) to vertices in V ($G_2$). When $G_1$ and $G_2$ are isomorphic, $G_1$M$G_2$ is just a permutation graph. In this work we derive
bounds for the k-restricted edge connectivity λ(k) of matched sum graphs $G_1$M$G_2$ for 2 ≤ k ≤ 5, and present some sufficient conditions for the optimality of λ(k)($G_1$M$G_2$).