http://hdl.handle.net/2099/10267 Facultat de Matemàtiques i Estadística, Universitat Politècnica de Catalunya, Barcelona, 9-11 June 2010 2013-06-19T15:38:04Z 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. 2011-01-01T00:00:00Z 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. 2011-01-01T00:00:00Z 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. 2011-01-01T00:00:00Z 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. 2011-01-01T00:00:00Z 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. 2011-01-01T00:00:00Z 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. 2011-01-01T00:00:00Z 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. 2011-01-01T00:00:00Z 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. 2011-01-01T00:00:00Z 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. 2011-01-01T00:00:00Z 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$G_2$). 2011-01-01T00:00:00Z 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. 2011-01-01T00:00:00Z 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. 2011-01-01T00:00:00Z 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. 2011-01-01T00:00:00Z 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. 2011-01-01T00:00:00Z 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. 2011-01-01T00:00:00Z