The maximal spectral radius of a digraph with (m+1)^2 - s edges

It is known that the spectral radius of a digraph with k edges is \le \sqrt{k}, and that this inequality is strict except when k is a perfect square. For k=m^2 + \ell, \ell fixed, m large, Friedland showed that the optimal digraph is obtained from the complete digraph on m vertices by adding one extra vertex, and a corresponding loop, and then connecting it to the first \lfloor \ell/2\rfloor vertices by pairs of directed edges (this is for odd \ell, for even \ell we add one extra edge to the new vertex). Using a combinatorial reciprocity theorem by Gessel, and a classification by Backelin on the digraphs on s edges having a maximal number of walks of length two, we obtain the following result: for fixed 0< s \neq 4, k=(m+1)^2 - s, m large, the maximal spectral radius of a digraph with k edges is obtained by the digraph which is constructed from the complete digraph on m+1 vertices by removing the loop at the last vertex together with \lfloor s/2 \rfloor pairs of directed edges that connect to the last vertex (if s is even, remove an extra edge connecting to the last vertex).Comment: 11 pages, 9 eps figures. To be presented at the conference FPSAC03. Submitted to Electronic Journal of Linear Algebra. Keywords: Spectral radius, digraphs, 0-1 matrices, Perron-Frobenius theorem, number of walk

Gradient-prolongation commutativity and graph theory

This Note gives conditions that must be imposed to algebraic multilevel discretizations involving at the same time nodal and edge elements so that a gradient-prolongation commutativity condition will be satisfied; this condition is very important, since it characterizes the gradients of coarse nodal functions in the coarse edge function space. They will be expressed using graph theory and they provide techniques to compute approximation bases at each level.Comment: 6 page

Perturbation of eigenvalues of matrix pencils and optimal assignment problem

We consider a matrix pencil whose coefficients depend on a positive parameter $\epsilon$, and have asymptotic equivalents of the form $a\epsilon^A$ when $\epsilon$ goes to zero, where the leading coefficient $a$ is complex, and the leading exponent $A$ is real. We show that the asymptotic equivalent of every eigenvalue of the pencil can be determined generically from the asymptotic equivalents of the coefficients of the pencil. The generic leading exponents of the eigenvalues are the "eigenvalues" of a min-plus matrix pencil. The leading coefficients of the eigenvalues are the eigenvalues of auxiliary matrix pencils, constructed from certain optimal assignment problems.Comment: 8 page

Traçabilité d'exigences temporelles dans l'outil UML/SysML TTool

La démonstration proposée concerne la traçabilité d'exigences tout au long du cycle de développement d'un système temps-réel, potentiellement distribué. L'outil TTool, basé sur un profil UML2, permet de saisir les exigences au format SysML, puis de confronter, par utilisation de techniques de vérification formelle, ces exigences aux diagrammes UML du système

Markov chains in a Dirichlet Environment and hypergeometric integrals

The aim of this text is to establish some relations between Markov chains in Dirichlet Environments on directed graphs and certain hypergeometric integrals associated with a particular arrangement of hyperplanes. We deduce from these relations and the computation of the connexion obtained by moving one hyperplane of the arrangement some new relations on important functionals of the Markov chain.Comment: 6 pages, preliminary not

Traitement du Signal sur Graphe : Interprétation en termes de Filtre de l'Apprentissage Semi-Supervisé sur Graphe

National audienceNous montrons comment les outils de traitement du signal sur graphe peuvent dégager des notions de fréquences sur les graphes pour étudier des données portées par les nœud d'un graphe. Prenant l'exemple de l'apprentissage semi-supervisé, nous montrons alors qu'il peut s'interpréter comme le filtre d'un signal sur graphe