DSpace Collection: Facultat de Matemàtiques i Estadística, Universitat Politècnica de Catalunya, Barcelona, 9-11 June 2010
http://hdl.handle.net/2099/10267
Facultat de Matemàtiques i Estadística, Universitat Politècnica de Catalunya, Barcelona, 9-11 June 2010Mon, 02 Mar 2015 23:18:11 GMT2015-03-02T23:18:11Zwebmaster.bupc@upc.eduUniversitat Politècnica de Catalunya. Servei de Biblioteques i DocumentaciónoTopology of Cayley graphs applied to inverse additive problems
http://hdl.handle.net/2099/10390
Title: Topology of Cayley graphs applied to inverse additive problems
Authors: Hamidoune, Yahya Ould
Abstract: 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.Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/2099/103902011-01-01T00:00:00ZHamidoune, Yahya OuldnoWe 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
Title: Algebraic characterizations of bipartite distance-regular graphs
Authors: Fiol Mora, Miquel Àngel
Abstract: 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.Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/2099/103892011-01-01T00:00:00ZFiol Mora, Miquel ÀngelnoBipartite 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
Title: A mathematical model for dynamic memory networks
Authors: Fàbrega Canudas, José; Fiol Mora, Miquel Àngel; Serra Albó, Oriol; Andrés Yebra, José Luis
Abstract: 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.Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/2099/103882011-01-01T00:00:00ZFàbrega Canudas, José; Fiol Mora, Miquel Àngel; Serra Albó, Oriol; Andrés Yebra, José LuisnoThe 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
Title: Large Edge-non-vulnerable Graphs
Authors: Delorme, Charles
Abstract: 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.Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/2099/103872011-01-01T00:00:00ZDelorme, CharlesnoIn 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
Title: Dual concepts of almost distance-regularity and the spectral excess theorem
Authors: Dalfó Simó, Cristina; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest; Dam, Edwin R. van
Abstract: 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.Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/2099/103862011-01-01T00:00:00ZDalfó Simó, Cristina; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest; Dam, Edwin R. vannoGenerally 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
Title: Large digraphs of given diameter and degree from coverings
Authors: Ždímalová, Mária; Staneková, Lubica
Abstract: 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.Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/2099/103852011-01-01T00:00:00ZŽdímalová, Mária; Staneková, LubicanoWe 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
Title: Fiedler’s Clustering on m-dimensional Lattice Graphs
Authors: Trajanovski, Stojan; Mieghem, Piet Van
Abstract: 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.Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/2099/103842011-01-01T00:00:00ZTrajanovski, Stojan; Mieghem, Piet VannoWe 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
Title: Large graphs of diameter two and given degree
Authors: Širáň, Jozef; Siagiová, Jana; Ždímalová, Mária
Abstract: 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.Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/2099/103832011-01-01T00:00:00ZŠiráň, Jozef; Siagiová, Jana; Ždímalová, MárianoLet 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
Title: An overview of the degree/diameter problem for directed, undirected and mixed graphs
Authors: Miller, Mirka
Abstract: 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.Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/2099/103822011-01-01T00:00:00ZMiller, MirkanoA 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
Title: On the k-restricted edge-connectivity of matched sum graphs
Authors: Marcote Ordax, Francisco Javier
Abstract: 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$G_2$).Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/2099/103812011-01-01T00:00:00ZMarcote Ordax, Francisco JaviernoA 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$G_2$).Application layer multicast algorithm
http://hdl.handle.net/2099/10377
Title: Application layer multicast algorithm
Authors: Machado Sánchez, Sergio; Ozón Górriz, Francisco Javier
Abstract: This paper presents a multicast algorithm, called MSM-s, for point-to-multipoint transmissions. The algorithm, which has
complexity O(n2) in respect of the number n of nodes, is easy to implement and can actually be applied in other point-to multipoint systems such as distributed computing. We analyze the algorithm and we provide some upper and lower bounds for
the multicast time delay.Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/2099/103772011-01-01T00:00:00ZMachado Sánchez, Sergio; Ozón Górriz, Francisco JaviernoThis paper presents a multicast algorithm, called MSM-s, for point-to-multipoint transmissions. The algorithm, which has
complexity O(n2) in respect of the number n of nodes, is easy to implement and can actually be applied in other point-to multipoint systems such as distributed computing. We analyze the algorithm and we provide some upper and lower bounds for
the multicast time delay.Radially Moore graphs of radius three and large odd degree
http://hdl.handle.net/2099/10376
Title: Radially Moore graphs of radius three and large odd degree
Authors: López, Nacho; Gómez Martí, José
Abstract: Extremal graphs which are close related to Moore graphs have been defined in different ways. Radially Moore graphs are one of these examples of extremal graphs. Although it is proved that radially Moore graphs exist for radius two, the general problem remains open. Knor, and independently Exoo, gives
some constructions of these extremal graphs for radius three and small degrees. As far as we know, some few examples have been found for other small values of the degree and the radius.
Here, we consider the existence problem of radially Moore graphs of radius three. We use the generalized undirected de Bruijn
graphs to give a general construction of radially Moore graphs of radius three and large odd degree.Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/2099/103762011-01-01T00:00:00ZLópez, Nacho; Gómez Martí, JosénoExtremal graphs which are close related to Moore graphs have been defined in different ways. Radially Moore graphs are one of these examples of extremal graphs. Although it is proved that radially Moore graphs exist for radius two, the general problem remains open. Knor, and independently Exoo, gives
some constructions of these extremal graphs for radius three and small degrees. As far as we know, some few examples have been found for other small values of the degree and the radius.
Here, we consider the existence problem of radially Moore graphs of radius three. We use the generalized undirected de Bruijn
graphs to give a general construction of radially Moore graphs of radius three and large odd degree.On the diameter of random planar graphs
http://hdl.handle.net/2099/10375
Title: On the diameter of random planar graphs
Authors: Chapuy, Guillaume; Fraser, Simon; Fusy, Eric; Giménez, Omer; Noy Serrano, Marc
Abstract: We show that the diameter D(Gn) of a random labelled connected planar graph with n vertices is asymptotically almost surely of order $n^{1/4}$, in the sense that there exists a constant c > 0 such that $P(D(G_n) \in{} (n^{1/4-\in{}} , n^{1/4+\in{}}))$ ≥ 1 − exp(−n^{ce})for $\in{}$ small enough and n large enough (n ≥ $n_0$($\in{}$)). We prove
similar statements for rooted 2-connected and 3-connected maps and planar graphs.Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/2099/103752011-01-01T00:00:00ZChapuy, Guillaume; Fraser, Simon; Fusy, Eric; Giménez, Omer; Noy Serrano, MarcnoWe show that the diameter D(Gn) of a random labelled connected planar graph with n vertices is asymptotically almost surely of order $n^{1/4}$, in the sense that there exists a constant c > 0 such that $P(D(G_n) \in{} (n^{1/4-\in{}} , n^{1/4+\in{}}))$ ≥ 1 − exp(−n^{ce})for $\in{}$ small enough and n large enough (n ≥ $n_0$($\in{}$)). We prove
similar statements for rooted 2-connected and 3-connected maps and planar graphs.On the vulnerability of some families of graphs
http://hdl.handle.net/2099/10374
Title: On the vulnerability of some families of graphs
Authors: Moreno Casablanca, Rocío; Díanez, Ana R.; García-Vázquez, Pedro
Abstract: The toughness of a noncomplete graph G is defined as τ (G) = min{|S|/ω(G − S)}, where the minimum is taken over all cutsets S of vertices of G and ω(G − S) denotes the number of components of the resultant graph G − S by deletion of S. In this paper, we investigate the toughness of the corona of two connected graphs and obtain the exact value for the corona of two graphs belonging to some families as paths, cycles, wheels
or complete graphs. We also get an upper and a lower bounds for the toughness of the cartesian product of the complete graph $K_2$ with a predetermined graph G.Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/2099/103742011-01-01T00:00:00ZMoreno Casablanca, Rocío; Díanez, Ana R.; García-Vázquez, PedronoThe toughness of a noncomplete graph G is defined as τ (G) = min{|S|/ω(G − S)}, where the minimum is taken over all cutsets S of vertices of G and ω(G − S) denotes the number of components of the resultant graph G − S by deletion of S. In this paper, we investigate the toughness of the corona of two connected graphs and obtain the exact value for the corona of two graphs belonging to some families as paths, cycles, wheels
or complete graphs. We also get an upper and a lower bounds for the toughness of the cartesian product of the complete graph $K_2$ with a predetermined graph G.Symmetric L-graphs
http://hdl.handle.net/2099/10373
Title: Symmetric L-graphs
Authors: Camarero, Cristóbal; Martínez, Carmen; Beivide Palacio, Julio Ramón
Abstract: In this paper we characterize symmetric L-graphs, which are either Kronecker products of two cycles or Gaussian graphs.
Vertex symmetric networks have the property that the communication load is uniformly distributed on all the vertices so that
there is no point of congestion. A stronger notion of symmetry, edge symmetry, requires that every edge in the graph looks the
same. Such property ensures that the communication load is uniformly distributed over all the communication links, so that
there is no congestion at any link.Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/2099/103732011-01-01T00:00:00ZCamarero, Cristóbal; Martínez, Carmen; Beivide Palacio, Julio RamónnoIn this paper we characterize symmetric L-graphs, which are either Kronecker products of two cycles or Gaussian graphs.
Vertex symmetric networks have the property that the communication load is uniformly distributed on all the vertices so that
there is no point of congestion. A stronger notion of symmetry, edge symmetry, requires that every edge in the graph looks the
same. Such property ensures that the communication load is uniformly distributed over all the communication links, so that
there is no congestion at any link.