José Gómez

Large generalized cycles

Abstract A generalized cycle is a digraph whose set of vertices is partitioned in several parts that are cyclically ordered in such a way that the vertices in one part are adjacent only to vertices in the next part. The problems considered in this paper are: 1. 1. To find generalized cycles with given maximum out-degree and diameter that have large order. 2. 2. To find generalized cycles with small diameter for given values of their maximum out-degree and order. A bound is given for both problems. It is proved that the first bound can only be attained for small values of the diameter. We present two new families of generalized cycles that provide some solutions to these problems. These families are a generalization of the generalized de Bruijn and Kautz digraphs and the bipartite digraphs BD ( d , n ).

Asymptotically large (Δ,D)-graphs

Graphs with maximum degree @D, diameter D and orders greater than (@D/@a)^D, for a constant @a<2, are proved to exist for infinitely many values of @D and for D larger than a fixed value.

Graphs on alphabets as models for large interconnection networks

Abstract Graphs on alphabets are constructed by labelling the vertices with words on a given alphabet, and specifying a rule that relates pairs of different words to define the edges. They have proved to be quite suitable to model large interconnection networks since their structure usually provides efficient routing algorithms. The aim of this paper is to present several infinite families of such graphs with a large number of vertices for given values of their diameter and maximum degree.

Synergies between optical and physical variables in intercepting parabolic targets

Interception requires precise estimation of time-to-contact (TTC) information. A long-standing view posits that all relevant information for extracting TTC is available in the angular variables, which result from the projection of distal objects onto the retina. The different timing models rooted in this tradition have consequently relied on combining visual angle and its rate of expansion in different ways with tau being the most well-known solution for TTC. The generalization of these models to timing parabolic trajectories is not straightforward. For example, these different combinations rely on isotropic expansion and usually assume first-order information only, neglecting acceleration. As a consequence no optical formulations have been put forward so far to specify TTC of parabolic targets with enough accuracy. It is only recently that context-dependent physical variables have been shown to play an important role in TTC estimation. Known physical size and gravity can adequately explain observed data of linear and free-falling trajectories, respectively. Yet, a full timing model for specifying parabolic TTC has remained elusive. We here derive two formulations that specify TTC for parabolic ball trajectories. The first specification extends previous models in which known size is combined with thresholding visual angle or its rate of expansion to the case of fly balls. To efficiently use this model, observers need to recover the 3D radial velocity component of the trajectory which conveys the isotropic expansion. The second one uses knowledge of size and gravity combined with ball visual angle and elevation angle. Taking into account the noise due to sensory measurements, we simulate the expected performance of these models in terms of accuracy and precision. While the model that combines expansion information and size knowledge is more efficient during the late trajectory, the second one is shown to be efficient along all the flight.

On Moore bipartite digraphs: ON MOORE BIPARTITE DIGRAPHS

In the context of the degree/diameter problem for directed graphs, it is known that the number of vertices of a strongly connected bipartite digraph satisfies a Moore-like bound in terms of its diameter k and the maximum out-degrees (d1; d2) of its partite sets of vertices. It has been proved that, when d1d2 > 1, the digraphs attaining such a bound, called Moore bipartite digraphs, only exist when 2 k 4. This paper deals with

Radial Moore graphs of radius three

The maximum number of vertices in a graph of specified degree and diameter cannot exceed the Moore bound. Graphs achieving this bound are called Moore graphs. Because Moore graphs are so rare, researchers have considered various relaxations of the Moore graph constraints. Since the diameter of a Moore graph is equal to its radius, one can consider graphs in which the condition on the diameter is relaxed, by one, while the condition on the radius is maintained. Such graphs are called radial Moore graphs. It has previously been shown that radial Moore graphs exist for all degrees when the radius is two. In this paper, we extend this result to radius three. We also construct examples that settle the existence question for a few new cases, and summarize the state of knowledge on the problem.

ON LARGE VERTEX SYMMETRIC 2-REACHABLE DIGRAPHS

We are interested in the construction of the largest possible vertex symmetric digraphs with the property that between any two vertices there is a walk of length two (that is, they are 2-reachable). Other than a theoretical interest, these digraphs and their generalizations may be used as models of interconnection networks for implementing parallelism. In these systems many nodes are connected with relatively few links and short paths between them and each node may execute, without modifications, the same communication software. On the other hand, a message sent from any vertex reaches all vertices, including the sender, in exactly two steps. In this work we present families of vertex symmetric 2-reachable digraphs with order attaining the upper theoretical bound for any odd degree. Some constructions for even degree are also given.

Superfluous edges and exponential expansions of De Bruijn and Kautz graphs

A new way to expand De Bruijn and Kautz graphs is presented. It consists of deleting superfluous sets of edges (i.e., those whose removal does not increase the diameter) and adding new vertices and new edges preserving the maximum degree and the diameter. The number of vertices added to the Kautz graph, for a fixed maximum degree greater than four, is exponential on the diameter. Tables with lower bounds for the order of superfluous sets of edges and the number of vertices that can be added, are presented.

Unilaterally connected large digraphs and generalized cycles

Lower and upper bounds on the order of digraphs and generalized p-cycles with a specified maximum degree and unilateral diameter are given for generic values of the parameters. Infinite families of digraphs attaining the bounds asymptotically or even exactly are presented. In particular, optimal results are proved for bipartite digraphs (p = 2) and digraphs with unilateral diameter 3.

On the unilateral (Δ, D*)-problem

Large digraphs of a specified maximum degree and unilateral diameter are given for small values of these parameters. The constructions are based on different techniques such as voltage digraphs, digraph products, join of cycles, and vertex duplication. Finally, a table with the results is given.