Margarida Mitjana

The generalized hierarchical product of graphs

A generalization of both the hierarchical product and the Cartesian product of graphs is introduced and some of its properties are studied. We call it the generalized hierarchical product. In fact, the obtained graphs turn out to be subgraphs of the Cartesian product of the corresponding factors. Thus, some well-known properties of this product, such as a good connectivity, reduced mean distance, radius and diameter, simple routing algorithms and some optimal communication protocols, are inherited by the generalized hierarchical product. Besides some of these properties, in this paper we study the spectrum, the existence of Hamiltonian cycles, the chromatic number and index, and the connectivity of the generalized hierarchical product.

Weakly distance-regular digraphs

We introduce the concept of weakly distance-regular digraph and study some of its basic properties. In particular, the (standard) distance-regular digraphs, introduced by Damerell, turn out to be those weakly distance-regular digraphs which have a normal adjacency matrix. As happens in the case of distance-regular graphs, the study is greatly facilitated by a family of orthogonal polynomials called the distance polynomials. For instance, these polynomials are used to derive the spectrum of a weakly distance-regular digraph. Some examples of these, digraphs, such as the butterfly and the cycle prefix digraph which are interesting for their applications, are analyzed in the light of the developed theory. Also, some new constructions involving the line digraph and other techniques are presented.

The Spectra of Wrapped Butterfly Digraphs

The knowledge of the spectrum of a (di)graph is relevant for estimating some of its structural properties, which provide information on the topological and communication properties of the corresponding networks. Among these properties, we have, for instance, edge-expansion and node-expansion, bisection width, diameter, maximum cut, connectivity, and partitions. In this paper, we determine the complete spectra (eigenvalues and multiplicities) of wrapped butterfly digraphs.

Gossiping in chordal rings under the line model

This paper is devoted to the gossip (or all-to-all) problem in the chordal ring under the one-port line model. The line model assumes long distance calls between non-neighboring processors. In this sense, the line model is strongly related to circuit-switched networks, wormhole routing, optical networks supporting wavelength division multiplexing, ATM switching, and networks supporting connected mode routing protocols. Since the chordal rings are competitors of networks as meshes or tori because of their short diameter and bounded degree, it is of interest to ask whether they can support intensive communications (typically all-to-all) as efficiently as these networks. We propose polynomial algorithms to derive optimal or near-optimal gossip protocols in the chordal ring.

The inverses of some circulant matrices

We present here necessary and sufficient conditions for the invertibility of some circulant matrices that depend on three parameters and moreover, we explicitly compute the inverse. Our study also encompasses a wide class of circulant symmetric matrices. The techniques we use are related with the solution of boundary value problems associated to second order linear difference equations. Consequently, we reduce the computational cost of the problem. In particular, we recover the inverses of some well known circulant matrices whose coefficients are arithmetic or geometric sequences, Horadam numbers among others. We also characterize when a general symmetric, circulant and tridiagonal matrix is invertible and in this case, we compute explicitly its inverse.

A Spectral Study of the Manhattan Networks

The multidimensional Manhattan networks are a family of digraphs with many appealing properties, such as vertex symmetry (in fact they are Cayley digraphs), easy routing, Hamiltonicity, and modular structure. From the known structural properties of these digraphs, we fully determine their spectra, which always contain the spectra of hypercubes. In particular, in the standard (two-dimensional) case it is shown that their line digraph structure imposes the presence of the zero eigenvalue with a large multiplicity.

Polynomial-Time Algorithms for Minimum-Time Broadcast in Trees

AbstractThis paper addresses the minimum-time broadcast problem under several modes of the line model, i.e., when long-distance calls can be placed along paths in the network. It is well known that the minimum-time broadcast problem can be solved in polynomial time under the single-port edge-disjoint paths mode. However, it is equally well known that either relaxing the model to the all-port edge-disjoint paths mode, or constraining the model to the single-port vertex-disjoint paths mode, leads to NP-complete problems; and exact solutions have been derived for specific topologies only (e.g., hypercubes or tori). In this paper we present polynomial-time algorithms for minimum-time broadcast in trees. These algorithms are obtained by application of an original technique called the merging method , which can be applied in a larger context, for instance, to solve the multicast problem or to address the restricted regimen. The merging method requires solving the minimal contention-free matrix problem whose solution presents some interest on its own.

Distance–regular graphs having the M-property

We analyse when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is an M-matrix; that is, it has non-positive off-diagonal elements or, equivalently when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is also the combinatorial Laplacian of another network. When this occurs we say that the distance–regular graph has the M-property. We prove that only distance–regular graphs with diameter up to three can have the M-property and we give a characterization of the graphs that satisfy the M-property in terms of their intersection array. Moreover, we exhaustively analyse strongly regular graphs having the M-property and we give some families of distance–regular graphs with diameter three that satisfy the M-property. Roughly speaking, we prove that all distance–regular graphs with diameter one; about half of the strongly regular graphs; only some imprimitive distance–regular graphs with diameter three, and no distance–regular graphs with diameter greater than three, have the M-property. In addition, we conjecture that no primitive distance–regular graph with diameter three has the M-property.

Optical Routing of Uniform Instances in Tori

We consider the problem of routing uniform communication instances in switched optical tori that use the wavelength division multiplexing (or WDM) approach. A communication instance is called uniform if it consists exactly of all pairs of nodes in the graph whose distance is equal to one from a specified set S = (d1; d2,...,dk). We give bounds on the optimal load induced on an edge for any uniform instance in a torus Tn×n. When k = 1, we prove necessary and sufficient conditions on the value in S relative to n for the wavelength index to be equal to the load. When k ≥ 2, we show that for any set S, there exists an n0, such that for all n > n0, there is an optimal wavelength assignment for the communication instance specified by S on the torus Tn×n. We also show an approximation for the wavelength index for any S and n. Finally, we give some results for rectangular tori.

Broadcasting in cycle prefix digraphs

Cycle prefix digraphs are directed Cayley coset graphs that have been proposed as a model of interconnection networks for parallel architectures. In this paper we present new details concerning their structure that are used to design a communication scheme leading to upper bounds on their broadcast time. When the diameter is two, the digraphs are Kautz digraphs and in this case our algorithm improves the known upper bounds for their broadcast time and is indeed optimal for small values of the degree.