On Some Applications of Graph Theory, I

In a series of papers, of which the present one is Part 1, it is shown that solutions to a variety of problems in distance geometry, potential theory and theory of metric spaces are provided by appropriate applications of graph theoretic results. (c) 1972 Published by Elsevier B.V

On some applications of graph theory, I.

In a series of papers, of which the present one is Part I, it is shown that solutions to a variety of problems in distance geometry, potential theory and theory of metric spaces are provided by appropriate applications of graph theoretic results. © 1972

Turán-Ramsey theorems and simple asymptotically extremal structures

This paper is a continuation of [10], where P. Erdos, A. Hajnal, V. T. Sos. and E. Szemeredi investigated the following problem: Assume that a so called forbidden graph L and a function f(n) = o(n) are fixed. What is the maximum number of edges a graph G(n) on n vertices can have without containing L as a subgraph, and also without having more than f(n) independent vertices? This problem is motivated by the classical Turan and Ramsey theorems, and also by some applications of the Turin theorem to geometry, analysis (in particular, potential theory) [27 29], [11-13]. In this paper we are primarily interested in the following problem. Let (G(n)) be a graph sequence where G(n) has n vertices and the edges of G(n) are coloured by the colours chi1,...,chi(r), so that the subgraph of colour chi(nu) contains no complete subgraph K(pnu), (nu = 1,...,r). Further, assume that the size of any independent set in G(n) is o(n) (as n --> infinity). What is the maximum number of edges in G(n) under these conditions? One of the main results of this paper is the description of a procedure yielding relatively simple sequences of asymptotically extremal graphs for the problem. In a continuation of this paper we shall investigate the problem where instead of alpha(G(n)) = o(n) we assume the stronger condition that the maximum size of a K(p)-free induced subgraph of G(n) is o(n)

Métrologie des graphes de terrain, application à la construction de ressources lexicales et à la recherche d'information

This thesis is organized in two parts : the first part focuses on measures of similarity (or proximity) between vertices of a graph, the second part on clustering methods for bipartite graph. A new measure of similarity between vertices, based on short time random walks, is introduced. The main advantage of the method is that it is insensitive to the density of the graph. A broad state of the art of similarities between vertices is then proposed, as well as experimental comparisons of these measures. This is followed by the proposal of a robust method for comparing graphs sharing the same set of vertices. This measure is shown to be applicable to the comparison and merging of synonymy networks. Finally an application for the enrichment of lexical resources is presented. It consists in providing candidate synonyms on the basis of already existing links. In the second part, a parallel between formal concept analysis and clustering of bipartite graph is established. This parallel leads to the particular case where a partition of one of the vertex groups can be determined whereas there is no corresponding partition on the other group of vertices. A simple method that addresses this problem is proposed and evaluated. Finally, a system of automatic classification of search results (Kodex) is presented. This system is an application of previously seen clustering methods. An evaluation on a collection of two million web pages shows the benefits of the approach and also helps to understand some differences between clustering methods.Cette thèse s'organise en deux parties : une première partie s'intéresse aux mesures de similarité (ou de proximité) définies entre les sommets d'un graphe, une seconde aux méthodes de clustering de graphe biparti. Une nouvelle mesure de similarité entre sommets basée sur des marches aléatoires en temps courts est introduite. Cette méthode a l'avantage, en particulier, d'être insensible à la densité du graphe. Il est ensuite proposé un large état de l'art des similarités entre sommets, ainsi qu'une comparaison expérimentale de ces différentes mesures. Cette première partie se poursuit par la proposition d'une méthode robuste de comparaison de graphes partageant le même ensemble de sommets. Cette méthode est mise en application pour comparer et fusionner des graphes de synonymie. Enfin une application d'aide à la construction de ressources lexicales est présentée. Elle consiste à proposer de nouvelles relations de synonymie à partir de l'ensemble des relations de synonymie déjà existantes. Dans une seconde partie, un parallèle entre l'analyse formelle de concepts et le clustering de graphe biparti est établi. Ce parallèle conduit à l'étude d'un cas particulier pour lequel une partition d'un des groupes de sommets d'un graphe biparti peut-être déterminée alors qu'il n'existe pas de partitionnement correspondant sur l'autre type de sommets. Une méthode simple qui répond à ce problème est proposée et évaluée. Enfin Kodex, un système de classification automatique des résultats d'une recherche d'information est présenté. Ce système est une application en RI des méthodes de clustering vues précédemment. Une évaluation sur une collection de deux millions de pages web montre les avantages de l'approche et permet en outre de mieux comprendre certaines différences entre méthodes de clustering