The degree-number of vertices problem in Manhattan networks

Generally speaking, the aim of this work is to study the problem (Delta,N) (or the degree-number of vertices problem) for the case of a Manhattan digraph. A digraph is a network formed by vertices and directed edges called arcs (in the case of graphs the edges have no direction). The diameter of a graph is the minimum distance that exists between two of the farthest vertices. In the diameter of a digraph we must take into account that arcs have direction. A double-step digraph consists of N vertices and a set of arcs of the form (i,i+a) and (i,i+b), with a and b positive integers called 'steps', that is, there are connections from vertex i to vertices i+a and i+b (operations are modulo N). This digraph is denoted by G(N;a,b). A double-step graph G(N;+-a,+-b) consists of N vertices, but the edges are of the form (i,i+-a) and (i,i+-b), with steps a and b (positive integers), therefore, there are connections from vertex i to vertices i+a, i-a, i+b and i-b (mod N). In a Manhattan digraph, the arcs have directions like the ones of the streets and avenues of Manhattan (or l'Eixample in Barcelona), that is, if an arc goes to the right, the 'next one' goes to the left and if an arc goes upwards, the 'next one' goes downwards. The (Delta,N) problem consists in finding the minimum diameter of a graph or digraph given the number of vertices N and the maximum degree Delta. As this problem has been solved for the case of double-step graphs G(N;+-a,+-b), we expand these graphs transforming every vertex into a directed cycle of order 4 and every edge into two arcs in opposite directions, so that we obtain a Manhattan digraph M. In this work we find the relation between the steps of the double-step graph G(N;+-a,+-b) and the ones of the Manhattan digraph M. Moreover, we made a program that calculates the diameter of the so-called New Amsterdam digraph NA, related to the Manhattan digraph M, from the parameters of the original graph G(N;+-a,+-b).Català: En termes generals, l’objectiu d’aquest treball és estudiar el problema (o problema grau-nombre de vèrtexs) per al cas del digraf Manhattan. Un digraf és una xarxa constituïda per vèrtexs i per arestes dirigides anomenades arcs (en el cas de grafs, les arestes no tenen direcció). El diàmetre d’un graf és la mínima distància possible que hi ha entre dos dels vèrtexs més allunyats entre si. En el diàmetre d’un digraf hem de tenir en compte que els arcs tenen direcció. Un digraf de doble pas consta de vèrtexs i un conjunt d'arcs de la forma i , amb i enters positius anomenats “passos", és a dir, que existeixen enllaços des del vèrtex cap els vèrtexs i (les operacions s'han d'entendre sempre mòdul ). Aquest digraf es denota . Un graf de doble pas també consta de vèrtexs, però les arestes són de la forma i , amb passos i (enters positius), per tant, existeixen enllaços des del vèrtex cap els vèrtexs i (mod ) . En un digraf Manhattan els arcs tenen les direccions com les dels carrers i les avingudes de Manhattan (o de l'Eixample de Barcelona), és a dir, si un arc va cap a la dreta, el "següent" va cap a l'esquerra i si un arc va cap a dalt, el "següent" va cap a baix. El problema consisteix a trobar el diàmetre mínim d'un graf o digraf fixats el nombre de vèrtexs i el grau . Com que aquest problema ha estat resolt per al cas de grafs de doble pas , hem expandit aquests grafs transformant cada vèrtex en un cicle dirigit de 4 vèrtexs i cada aresta en dos arcs de sentits oposats, de manera que obtenim un digraf Manhattan . En aquest treball trobem la relació entre els passos del graf de doble pas i els del digraf Manhattan . A més, hem fet un programa que calcula el diàmetre del digraf anomenat New Amsterdam , que està relacionat amb el Manhattan , a partir dels paràmetres del graf original

The (Delta,D) and (Delta,N) problems in double-step digraphs with unilateral distance

We study the (delta;D) and (delta;N) problems for double-step digraphs considering the unilateral distance, which is the minimum between the distance in the digraph and the distance in its converse digraph, the latter obtained by changing the directions of all the arcs. The first problem consists of maximizing the number of vertices N of a digraph, given the maximum degree and the unilateral diameter D , whereas the second one (somehow dual of the first) consists of minimizing the unilateral diameter given the maximum degree and the number of vertices. We solve the first problem for every value of the unilateral diameter and the second one for infinitely many values of the number of vertices. Moreover, we compute the mean unilateral distance of the digraphs in the families considered.Postprint (published version

The (Delta,D) and (Delta,N) problems in double-step digraphs with unilateral distance

We study the (delta;D) and (delta;N) problems for double-step digraphs considering the unilateral distance, which is the minimum between the distance in the digraph and the distance in its converse digraph, the latter obtained by changing the directions of all the arcs. The first problem consists of maximizing the number of vertices N of a digraph, given the maximum degree and the unilateral diameter D , whereas the second one (somehow dual of the first) consists of minimizing the unilateral diameter given the maximum degree and the number of vertices. We solve the first problem for every value of the unilateral diameter and the second one for infinitely many values of the number of vertices. Moreover, we compute the mean unilateral distance of the digraphs in the families considered

From spline wavelet to sampling theory on circulant graphs and beyond– conceiving sparsity in graph signal processing

Graph Signal Processing (GSP), as the field concerned with the extension of classical signal processing concepts to the graph domain, is still at the beginning on the path toward providing a generalized theory of signal processing. As such, this thesis aspires to conceive the theory of sparse representations on graphs by traversing the cornerstones of wavelet and sampling theory on graphs. Beginning with the novel topic of graph spline wavelet theory, we introduce families of spline and e-spline wavelets, and associated filterbanks on circulant graphs, which lever- age an inherent vanishing moment property of circulant graph Laplacian matrices (and their parameterized generalizations), for the reproduction and annihilation of (exponen- tial) polynomial signals. Further, these families are shown to provide a stepping stone to generalized graph wavelet designs with adaptive (annihilation) properties. Circulant graphs, which serve as building blocks, facilitate intuitively equivalent signal processing concepts and operations, such that insights can be leveraged for and extended to more complex scenarios, including arbitrary undirected graphs, time-varying graphs, as well as associated signals with space- and time-variant properties, all the while retaining the focus on inducing sparse representations. Further, we shift from sparsity-inducing to sparsity-leveraging theory and present a novel sampling and graph coarsening framework for (wavelet-)sparse graph signals, inspired by Finite Rate of Innovation (FRI) theory and directly building upon (graph) spline wavelet theory. At its core, the introduced Graph-FRI-framework states that any K-sparse signal residing on the vertices of a circulant graph can be sampled and perfectly reconstructed from its dimensionality-reduced graph spectral representation of minimum size 2K, while the structure of an associated coarsened graph is simultaneously inferred. Extensions to arbitrary graphs can be enforced via suitable approximation schemes. Eventually, gained insights are unified in a graph-based image approximation framework which further leverages graph partitioning and re-labelling techniques for a maximally sparse graph wavelet representation.Open Acces