The work in this thesis concerns the combinatorial theory of graphs, algebraic combinatorics and ... more The work in this thesis concerns the combinatorial theory of graphs, algebraic combinatorics and discrete geometry. On one side, it is about enumerating Hamiltonian paths and cycles of a given type in a tournament; On the other side, it studies numerical sequences verifying a quadratic difference equation.Concerning the results of the first part, we find: an equality between the number of Hamiltonians paths (resp. cycles) of a given type, in a tournament and its complement; an expression of the number of Hamiltonian oriented paths of a given type in a transitive tournament in terms of a recursive function F called the « path-function »; and the construction of an algorithm to compute F.In the second part of the work, we study cyclic graphs altogether with a solution to a quadratic difference equation.A parameter of this equation distinguishes real and complex sequences. A correspondence between real solutions and a class of polynomials with positive integer coefficients is establish...
We construct a combinatorial function F which computes the number of oriented Hamiltonian paths o... more We construct a combinatorial function F which computes the number of oriented Hamiltonian paths of any given type, in a transitive tournament. We also study many properties of F that arise, and reach some observations.
Université de Bretagne occidentale - Brest ; Université libanaise, Jan 31, 2020
Le travail de cette thèse se situe dans les domaines de la théorie combinatoire des graphes, la c... more Le travail de cette thèse se situe dans les domaines de la théorie combinatoire des graphes, la combinatoire algébrique et la géométrie discrète. D'un part, il concerne l'énumération des chemins et cycles Hamiltoniens de type donné dans un tournoi ; de l'autre part, il étudie des suites numériques vérifiant une équation à différence quadratique. Parmi les résultats obtenus dans la première partie, on trouve : une égalité entre le nombre des chemins (resp. cycles) Hamiltoniens d’un type donné dans un tournoi et dans son complément; une expression du nombre de chemins Hamiltoniens d’un type donné pour un tournoi transitif en termes d'une fonction récursive F appelée « path-function »; la construction d'un algorithme pour le calcul de F. L'objet fondamental dans la deuxième partie est un graphe cyclique muni d'une solution d'une équation à différence quadratique. Un paramètre de cette équation distingue les solutions réelles et les solutions complexes. Une correspondance entre les solutions réelles et une classe de polynômes à coefficients entiers positifs est établie. Pour compléter la correspondance, les digraphes Eulériens à un pas interviennent. Une solution complexe détermine une marche fermée dans le plan pour laquelle à chaque pas on tourne à gauche ou à droite par un angle constant (l'angle tournant). Cette fois-ci les polynômes cyclotomiques jouent un rôle important. La caractérisation des polynômes qui déterminent de telles suites est un problème qu’on surmonte afin d'élucider des propriétés géométriques de tels cycles polygonaux. Notamment, lorsque la marche exploite les côtés d'un polygone régulier avec angle extérieur 2π/n, on trouve des phénomènes non anticipés lorsque n≥12.The work in this thesis concerns the combinatorial theory of graphs, algebraic combinatorics and discrete geometry. On one side, it is about enumerating Hamiltonian paths and cycles of a given type in a tournament; On the other side, it studies numerical sequences verifying a quadratic difference equation.Concerning the results of the first part, we find: an equality between the number of Hamiltonians paths (resp. cycles) of a given type, in a tournament and its complement; an expression of the number of Hamiltonian oriented paths of a given type in a transitive tournament in terms of a recursive function F called the « path-function »; and the construction of an algorithm to compute F.In the second part of the work, we study cyclic graphs altogether with a solution to a quadratic difference equation.A parameter of this equation distinguishes real and complex sequences. A correspondence between real solutions and a class of polynomials with positive integer coefficients is established. To complete the correspondence, 1-step Eulerian digraphs interfere. A complex solution determines a closed planar walk in the plane, for which at each step we turn either left or right by a constant angle (the turning angle). This time, cyclotomic polynomials play a major role. Characterizing polynomials that determine such a solution is a problem that we study to the end of finding geometric properties of such polygonal cycles.When the walk exploits the sides of a regular polygon with exterior angle 2 π/n, we find unexpected phenomena when n≥ 12
We prove that a tournament T and its complement T contain the same number of oriented Hamiltonian... more We prove that a tournament T and its complement T contain the same number of oriented Hamiltonian paths (resp. cycles) of any given type, as a generalization of Rosenfeld's result proved for antidirected paths.
We explore relations between cyclic sequences determined by a quadratic difference relation, cycl... more We explore relations between cyclic sequences determined by a quadratic difference relation, cyclotomic polynomials, Eulerian digraphs and walks in the plane. These walks correspond to closed paths for which at each step one must turn either left or right through a fixed angle. In the case when this angle is 2π/n, then non-symmetric phenomena occurs for n ≥ 12. Examples arise from algebraic numbers of modulus one which are not n'th roots of unity. In particular, for there to exist a non-constant real solution to (1), necessarily γ ≥ 1. Equally, if γ < 1, then for any three successive terms, at least one must be complex. An example of an integer QCS of order 10 is given by (4)
The work in this thesis concerns the combinatorial theory of graphs, algebraic combinatorics and ... more The work in this thesis concerns the combinatorial theory of graphs, algebraic combinatorics and discrete geometry. On one side, it is about enumerating Hamiltonian paths and cycles of a given type in a tournament; On the other side, it studies numerical sequences verifying a quadratic difference equation.Concerning the results of the first part, we find: an equality between the number of Hamiltonians paths (resp. cycles) of a given type, in a tournament and its complement; an expression of the number of Hamiltonian oriented paths of a given type in a transitive tournament in terms of a recursive function F called the « path-function »; and the construction of an algorithm to compute F.In the second part of the work, we study cyclic graphs altogether with a solution to a quadratic difference equation.A parameter of this equation distinguishes real and complex sequences. A correspondence between real solutions and a class of polynomials with positive integer coefficients is establish...
We construct a combinatorial function F which computes the number of oriented Hamiltonian paths o... more We construct a combinatorial function F which computes the number of oriented Hamiltonian paths of any given type, in a transitive tournament. We also study many properties of F that arise, and reach some observations.
Université de Bretagne occidentale - Brest ; Université libanaise, Jan 31, 2020
Le travail de cette thèse se situe dans les domaines de la théorie combinatoire des graphes, la c... more Le travail de cette thèse se situe dans les domaines de la théorie combinatoire des graphes, la combinatoire algébrique et la géométrie discrète. D'un part, il concerne l'énumération des chemins et cycles Hamiltoniens de type donné dans un tournoi ; de l'autre part, il étudie des suites numériques vérifiant une équation à différence quadratique. Parmi les résultats obtenus dans la première partie, on trouve : une égalité entre le nombre des chemins (resp. cycles) Hamiltoniens d’un type donné dans un tournoi et dans son complément; une expression du nombre de chemins Hamiltoniens d’un type donné pour un tournoi transitif en termes d'une fonction récursive F appelée « path-function »; la construction d'un algorithme pour le calcul de F. L'objet fondamental dans la deuxième partie est un graphe cyclique muni d'une solution d'une équation à différence quadratique. Un paramètre de cette équation distingue les solutions réelles et les solutions complexes. Une correspondance entre les solutions réelles et une classe de polynômes à coefficients entiers positifs est établie. Pour compléter la correspondance, les digraphes Eulériens à un pas interviennent. Une solution complexe détermine une marche fermée dans le plan pour laquelle à chaque pas on tourne à gauche ou à droite par un angle constant (l'angle tournant). Cette fois-ci les polynômes cyclotomiques jouent un rôle important. La caractérisation des polynômes qui déterminent de telles suites est un problème qu’on surmonte afin d'élucider des propriétés géométriques de tels cycles polygonaux. Notamment, lorsque la marche exploite les côtés d'un polygone régulier avec angle extérieur 2π/n, on trouve des phénomènes non anticipés lorsque n≥12.The work in this thesis concerns the combinatorial theory of graphs, algebraic combinatorics and discrete geometry. On one side, it is about enumerating Hamiltonian paths and cycles of a given type in a tournament; On the other side, it studies numerical sequences verifying a quadratic difference equation.Concerning the results of the first part, we find: an equality between the number of Hamiltonians paths (resp. cycles) of a given type, in a tournament and its complement; an expression of the number of Hamiltonian oriented paths of a given type in a transitive tournament in terms of a recursive function F called the « path-function »; and the construction of an algorithm to compute F.In the second part of the work, we study cyclic graphs altogether with a solution to a quadratic difference equation.A parameter of this equation distinguishes real and complex sequences. A correspondence between real solutions and a class of polynomials with positive integer coefficients is established. To complete the correspondence, 1-step Eulerian digraphs interfere. A complex solution determines a closed planar walk in the plane, for which at each step we turn either left or right by a constant angle (the turning angle). This time, cyclotomic polynomials play a major role. Characterizing polynomials that determine such a solution is a problem that we study to the end of finding geometric properties of such polygonal cycles.When the walk exploits the sides of a regular polygon with exterior angle 2 π/n, we find unexpected phenomena when n≥ 12
We prove that a tournament T and its complement T contain the same number of oriented Hamiltonian... more We prove that a tournament T and its complement T contain the same number of oriented Hamiltonian paths (resp. cycles) of any given type, as a generalization of Rosenfeld's result proved for antidirected paths.
We explore relations between cyclic sequences determined by a quadratic difference relation, cycl... more We explore relations between cyclic sequences determined by a quadratic difference relation, cyclotomic polynomials, Eulerian digraphs and walks in the plane. These walks correspond to closed paths for which at each step one must turn either left or right through a fixed angle. In the case when this angle is 2π/n, then non-symmetric phenomena occurs for n ≥ 12. Examples arise from algebraic numbers of modulus one which are not n'th roots of unity. In particular, for there to exist a non-constant real solution to (1), necessarily γ ≥ 1. Equally, if γ < 1, then for any three successive terms, at least one must be complex. An example of an integer QCS of order 10 is given by (4)
Uploads
Papers by Zeina Hanna