Dans cette thèse, nous nous concentrons sur l’étude des algèbres de Hopf de type I, à savoir de type sélection/quotient. Nous présentons une structure d’algèbre de Hopf sur l’espace vectoriel engendré par les mots tassés avec du coproduit sélection/quotient. C’est un algèbre libre sur ses mots irreductible. Nous montrons que la série de Hilbert de cette algèbre de Hopf. Nous donnons une nouvelle preuve de l’universalité du polynôme de Tutte pour les matroïdes.Cette preuve utilise des caractères appropriés de l’algèbre de Hopf des matroïdes introduite par Schmitt (1994). Nous montrons que ces caractères sont des solutions des équations différentielles du même type que les équations différentielles utilisées pour décrire le flux du groupe de renormalisation en théorie quantique de champs. Cette approche nous permet aussi de démontrer,d’une manière différente, une formule de convolution du polynôme de Tutte des matroïdes,formule publiée par Kook, Reiner et Stanton (1999) et par Etienne et Las Vergnas (1998). Dans la dernière partie, nous définissons une algèbre de Hopf non-commutative de graphes. Lanon-commutativité du produit est obtenue grâce à des étiquettes entières distinctes associées aux arrêtes du graphe. Cette idée est inspirée de certaines techniques analytiques utilisées en renormalisation en théories quantiques des champs. Nous définissons ensuite une structure d’algèbre de Hopf, avec un coproduit basé sur une règle de type sélection/quotient, et nous démontrons la coassociativité de ce coproduit. Nous analysons finalement la structure de quadri-cogèbre et les structures codendriformes associées. / In this thesis, we focus on the study of Hopf algebras of type I, namely the selection/quotient.We study the new Hopf algebra structure on the vector space spanned by packed words. Weshow that this algebra is free on its irreducible packed words. We also compute the Hilbertseries of this Hopf algebra.We provide a new way to obtain the universality of the Tutte polynomial for matroids. Thisproof uses appropriate characters of Hopf algebra of matroids, algebra introduced by Schmitt(1994). We show that these Hopf algebra characters are solutions of some differential equationswhich are of the same type as the differential equations used to describe the renormalizationgroup flow in quantum field theory. This approach allows us to also prove, in a different way, amatroid Tutte polynomial convolution formula published by Kook, Reiner and Stanton (1999)and by Etienne and Las Vergnas (1998).We define a non-commutative Hopf algebra of graphs. The non-commutativity of the productis obtained thanks to some discrete labels associated to the graph edges. This idea is inspiredfrom certain analytic techniques used in quantum field theory renormalization. We then definea Hopf algebra structure, with a coproduct based on a selection/quotient rule, and prove thecoassociativity of this coproduct. We analyze the associated quadri-coalgebra and codendrifromstructures.

We are studying the Tutte Polynomial of hyperplane arrangements. We discuss some previous work done to compute these polynomials. Then we explain our method to calculate the Tutte Polynomial of some arrangements more efficiently. We next discuss the details of the program used to do the calculation. We use this program and present the actual Tutte Polynomials calculated for the arrangements E6, E7, and E8. / text

This thesis consists of three self-contained but related parts. The rst is focussed on polymatroids, these being a natural generalisation of matroids. The Tutte polynomial is one of the most important and well-known graph polynomials, and also features prominently in matroid theory. It is however not directly applicable to polymatroids. For instance, deletion-contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. The second section is concerned with split matroids, a class of matroids which arises by putting conditions on the system of split hyperplanes of the matroid base polytope. We describe these conditions in terms of structural properties of the matroid, and use this to give an excluded minor characterisation of the class. In the nal section, we investigate the structure of clutters. A clutter consists of a nite set and a collection of pairwise incomparable subsets. Clutters are natural generalisations of matroids, and they have similar operations of deletion and contraction. We introduce a notion of connectivity for clutters that generalises that of connectivity for matroids. We prove a splitter theorem for connected clutters that has the splitter theorem for connected matroids as a special case: if M and N are connected clutters, and N is a proper minor of M, then there is an element in E(M) that can be deleted or contracted to produce a connected clutter with N as a minor.

A Tutte trail T of a graph G is a trail such that every component of GnV (T) has at most three edges connecting it to T. In 1992, Bill Jackson conjectured that every 2-edge-connected graph G has a Tutte closed trail. In this thesis, we show that Jackson's conjecture is true when G is embedded on the plane and the projective plane. We also give some partial results when G is embedded on the torus.

L'objectif visé dans ce travail consiste en la présentation du polynôme de Tutte, et ce à la manière de son idéateur, M. William Thomas Tutte. Nous dressons également un portrait de l'éventail des applications possibles de ce polynôme, notamment en théorie des graphes, en physique de la mécanique statistique, de même qu'en théorie des noeuds. À cet égard, nous faisons la démonstration que le polynôme de Tutte admet une spécialisation en terme de la fonction de partition d'un modèle de Potts, ainsi qu'en terme du polynôme de Jones d'un entrelacs alterné. Ce travail se conclut par une série de calculs sur les graphes 2-connexes et connexes, pour lesquels nous utilisons une équation fonctionnelle bien connue de la théorie des espèces, de même que des fonctions de poids bloc-multiplicatives. Ces calculs nous ont permis, entre autres, d'établir l'égalité entre le poids total des λ-flots à flux non nuls sur les graphes 2-connexes à quatre sommets et le nombre de marelles de longueur trois dans l'hypercube de dimension λ -1. ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : Polynôme de Tutte, Polynôme chromatique, Polynôme de flot, Polynôme de fiabilité, Polynôme de Jones, Entrelacs alterné, Fonction de partition, Modèle de Potts, Graphes 2-connexes.

Under the leadership of the King Joseph II, various musical activities were flourishing in Vienna during the late 18th century. Among all, the opera buffa was one of the major forms of musical performances. During the stay in Vienna, Mozart and Theater Poet Da Ponte composed three opere buffe together, and the last one was Così fan tutte composed in 1790. Among all the opere buffe performed in Vienna from 1783 to 1791, the ratio of ensembles to arias in Così fan tutte was the highest. Therefore, the purpose of this thesis is to study the ensembles in this work, analyze the musical characterization of all characters, and carefully examine the dramatic meaning from the musical structure. In this thesis, there are three chapters except the preface and conclusion. The first chapter introduces the opere buffe performed in Vienna from 1783 to 1791 and describes its social meaning at the time, which provides the background as Mozart composed his opere buffe. The second chapter discusses the origin of the libretto and the influence of the Commedia dell'arte to the characteristic structure. This chapter therefore expresses its ideas more distinctly about two origins of the major play, the historical basis of six characters¡¦ temperaments, and Da Ponte¡¦s inductive composition technique. In addition, the third chapter further probes the musical intention of the ensembles in Così fan tutte. Furnishing characters in this play with vivid portraits and energetic vitality, Mozart composed this opera by innovative music language and various structure. Così fan tutte was harshly disparaged by critics in 19th century and being neglected for almost one century. Nevertheless, it was being highly recognized as moral standard was gradually dropped. It is fairly stated that Così fan tutte is a major achievement of opera buffa in late 18th century for Mozart¡¦s overflowing creativity in composing ensembles as well as his keen penetration for drama.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-09T18:02:46Z No. of bitstreams: 1 LINO MARCOS DA SILVA - DISSERTAÇÃO PPGMAT 2006..pdf: 606293 bytes, checksum: f6729428e1a4d16d1b38704fe9b418a4 (MD5) / Made available in DSpace on 2018-07-09T18:02:46Z (GMT). No. of bitstreams: 1 LINO MARCOS DA SILVA - DISSERTAÇÃO PPGMAT 2006..pdf: 606293 bytes, checksum: f6729428e1a4d16d1b38704fe9b418a4 (MD5) Previous issue date: 2006-03 / Capes / Neste trabalho apresentamos algumas relações entre matróides e códigos lineares. Estudamos vários invariantes numéricos de matróides e vemos que este é um dos muitos aspectos de teoria das matróides que tiveram origem em teoria dos grafos. Analisamos uma classe especial de tais invariantes: os invariantes Tutte-Grothendieck. Mostramos que o polinômio de Tutte é o invariante T-Guniversal (Brilawski,1972) e o relacionamos à teoria dos códigos mostrando que a distribuição de pesos de palavras-código em um código linear é um invariante T-G generalizado (Greene,1976). / In this work we present a relation between matroid and linear codes. Numericals invariants for matroids is one the many topics of matroid theory having its origins graph theory. The Tutte Polynomial of the matroid play a role very important in various problems concerned with such invariants. In 1972 Brylawski showed that the Tutte Polynomial is a T-G invariant. In 1976, Greene established a relation among linear codes and the Tutte Polynomial showing that the distribuition of codeweigths in a linear codes is a generalized T-G invariant.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-09T17:32:41Z No. of bitstreams: 1 MARTA ÉLID CONCEIÇÃO AMORIM - DISSERTAÇÃO PPGMAT 2006..pdf: 311493 bytes, checksum: 839fa55790d058f7e88a3735ab098065 (MD5) / Made available in DSpace on 2018-07-09T17:32:41Z (GMT). No. of bitstreams: 1 MARTA ÉLID CONCEIÇÃO AMORIM - DISSERTAÇÃO PPGMAT 2006..pdf: 311493 bytes, checksum: 839fa55790d058f7e88a3735ab098065 (MD5) Previous issue date: 2006-02 / Capes / Neste trabalho apresentamos o Polinômio de Tutte com duas e quatro variáveis e o associamos a Função de Möbius, o Polinômio Característico e o Beta Invariante. Veremos, também, que por recursão sobre certas equações, algumas funções relacionadas a matróides podem ser obtidas pela avaliação do Polinômio de Tutte para certos valores. / Inthiswork,weshowTuttePolynomialwithtwoorfourvariablesandweassociateitwith Möbius Function the charateristic Polynomial and the Beta Invariant. We also see, to turn to undersomeequations,somefunctionsrelatedtomatroidscanbeobtainedforTuttePolynomial assessmentforsomevalues.

Cette thèse porte sur le polynôme de Tutte, étudié selon différents points de vue. Dans une première partie, nous nous intéressons à l’énumération des cartes planaires munies d’une forêt couvrante, ici appelées cartes forestières, avec un poids z par face et un poids u par composante non racine de la forêt. De manière équivalente, nous comptons selon le nombre de faces les cartes planaires C pondérées par TC(u + 1; 1), où TC désigne le polynôme de Tutte de C. Nous commençons par une caractérisation purement combinatoire de la série génératrice correspondante, notée F(z; u). Nous en déduisons que F(z; u) est différentiellement algébrique en z, c’est-à-dire que F satisfait une équation différentielle polynomiale selon z. Enfin, pour u ≥ -1, nous étudions le comportement asymptotique du n-ième coefficient de F(z; u). Nous observons une transition de phase en 0, avec notamment un régime très atypique en n-3 ln-2(n) pour u ϵ [-1; 0[, témoignant d’une nouvelle classe d’universalité pour les cartes planaires. Dans une seconde partie, nous proposons un cadre unificateur pour les différentes notions d’activités utilisées dans la littérature pour décrire le polynôme de Tutte.La nouvelle notion d’activité ainsi définie est appelée Δ-activité. Elle regroupe toutes les notions d’activité déjà connues et présente de belles propriétés, comme celle de Crapo qui définit une partition (adaptée à l’activité) du treillis des sous-graphes couvrants en intervalles. Nous conjecturons en dernier lieu que toute activité qui décrit le polynôme de Tutte et qui satisfait la propriété susmentionnée de Crapo peut être définie en termes de Δ-activités. / This thesis deals with the Tutte polynomial, studied from different points of view. In the first part, we address the enumeration of planar maps equipped with a spanning forest, here called forested maps, with a weight z per face and a weight u per non-root component of the forest. Equivalently, we count (with respect to the number of faces) the planar maps C weighted by TC(u + 1; 1), where TC is the Tutte polynomial of C.We begin by a purely combinatorial characterization of the corresponding generating function, denoted by F(z; u). We deduce from this that F(z; u) is differentially algebraic in z, that is, satisfies a polynomial differential equation in z. Finally, for u ≥ -1, we study the asymptotic behaviour of the nth coefficient of F(z; u).We observe a phase transition at 0, with a very unusual regime in n-3 ln-2(n) for u ϵ [-1; 0[, which testifiesa new universality class for planar maps. In the second part, we propose a framework unifying the notions of activity used in the literature to describe the Tutte polynomial. The new notion of activity thereby defined is called Δ-activity. It gathers all the notions of activities that were already known and has nice properties, as Crapo’s property that defines a partition of the lattice of the spanning subgraphs into intervals with respect to the activity. Lastly we conjecture that every activity that describes the Tutte polynomial and that satisfies Crapo’s property can be defined in terms of Δ-activity.

We begin by introducing matroids in the context of finite collections of vectors from a vector space over a specified field, where the notion of independence is linear independence. Then we will introduce the concept of a matroid invariant. Specifically, we will look at the Tutte polynomial, which is a well-defined two-variable invariant that can be used to determine differences and similarities between a collection of given matroids. The Tutte polynomial can tell us certain properties of a given matroid (such as the number of bases, independent sets, etc.) without the need to manually solve for them. Although the Tutte polynomial gives us significant information about a matroid, it does not uniquely determine a matroid. This thesis will focus on non-isomorphic matroids that have the same Tutte polynomial. We call such matroids Tutte-equivalent, and we will study the characteristics needed for two matroids to be Tutte-equivalent. Finally, we will demonstrate methods to construct families of Tutte-equivalent matroids.