The Regularity Lemma of Szemeredi is a fundamental tool in extremal graph theory with a wide range of applications in theoretical computer science. Partly as a recognition of his work on the Regularity Lemma, Endre Szemeredi has won the Abel Prize in 2012 for his outstanding achievement. In this thesis we present both practical and theoretical applications of the Regularity Lemma. The practical applications are concerning the important problem of data clustering, the theoretical applications are concerning the monochromatic vertex partition problem. In spite of its numerous applications to establish theoretical results, the Regularity Lemma has a drawback that it requires the graphs under consideration to be astronomically large, thus limiting its practical utility. As stated by Gowers, it has been ``well beyond the realms of any practical applications', the existing applications have been theoretical, mathematical. In the first part of the thesis, we propose to change this and we propose some modifications to the constructive versions of the Regularity Lemma. While this affects the generality of the result, it also makes it more useful for much smaller graphs. We call this result the practical regularity partitioning algorithm and the resulting clustering technique Regularity Clustering. This is the first integrated attempt in order to make the Regularity Lemma applicable in practice. We present results on applying regularity clustering on a number of benchmark data-sets and compare the results with k-means clustering and spectral clustering. Finally we demonstrate its application in Educational Data Mining to improve the student performance prediction. In the second part of the thesis, we study the monochromatic vertex partition problem. To begin we briefly review some related topics and several proof techniques that are central to our results, including the greedy and absorbing procedures. We also review some of the current best results before presenting ours, where the Regularity Lemma has played a critical role. Before concluding we discuss some future research directions that appear particularly promising based on our work.

Teste de propriedades em grafos consiste no estudo de algoritmos aleatórios sublineares que determinam se um grafo $G$ de entrada com $n$ vértices satisfaz uma dada propriedade ou se é necessário adicionar ou remover mais do que $\\epsilon{n \\choose 2}$ arestas para fazer $G$ satisfazê-la, para algum parâmetro $\\epsilon$ de erro fixo. Uma propriedade de grafos $P$ é dita testável se, para todo $\\epsilon > 0$, existe um tal algoritmo para $P$ cujo tempo de execução é independente de $n$. Um dos resultados de maior importância nesta área, provado por Alon e Shapira, afirma que toda propriedade hereditária de grafos é testável. Neste trabalho, apresentamos resultados análogos para torneios --- grafos completos nos quais são dadas orientações para cada aresta. / Graph property testing is the study of randomized sublinear algorithms which decide if an input graph $G$ with $n$ vertices satisfies a given property or if it is necessary to add or remove more than $\\epsilon{n \\choose 2}$ edges to make $G$ satisfy it, for some fixed error parameter $\\epsilon$ . A graph property $P$ is called testable if, for every $\\epsilon > 0$, there is such an algorithm for $P$ whose run time is independent of $n$. One of the most important results in this area is due to Alon and Shapira, who showed that every hereditary graph property is testable. In this work, we show analogous results for tournaments --- complete graphs in which every edge is given an orientation.

In this thesis, we present analytical analysis of key distribution schemes on wireless sensor networks. Since wireless sensor network is under unreliable environment, many random key pre-distribution based schemes have been developed to enhance security. Most of these schemes need to guarantee the existence of specific properties, such as disjoint secure paths or disjoint secure cliques, to achieve a secure cooperation among nodes. Two of the basic questions are as follows: 1. Under what conditions does a large-scale sensor network contain a certain structure? 2. How can one give a quantitative analysis behave as n grows to the infinity? However, analyzing such a structure or combinatorial problem is complicated in classical wireless network models such as percolation theories or random geometric graphs. Particularly, proofs in geometric setting models often blend stochastic geometric and combinatorial techniques and are more technically challenging. To overcome this problem, an approximative quasi-random graph is employed to eliminate some properties that are difficult to tackle. The most well-known solutions of this kind problems are probably Szemeredi's regularity lemma for embedding. The main difficulty from the fact that the above questions involve extremely small probabilities. These probabilities are too small to estimate by means of classical tools from probability theory, and thus a specific counting methods is inevitable.

Teste de propriedades em grafos consiste no estudo de algoritmos aleatórios sublineares que determinam se um grafo $G$ de entrada com $n$ vértices satisfaz uma dada propriedade ou se é necessário adicionar ou remover mais do que $\\epsilon{n \\choose 2}$ arestas para fazer $G$ satisfazê-la, para algum parâmetro $\\epsilon$ de erro fixo. Uma propriedade de grafos $P$ é dita testável se, para todo $\\epsilon > 0$, existe um tal algoritmo para $P$ cujo tempo de execução é independente de $n$. Um dos resultados de maior importância nesta área, provado por Alon e Shapira, afirma que toda propriedade hereditária de grafos é testável. Neste trabalho, apresentamos resultados análogos para torneios --- grafos completos nos quais são dadas orientações para cada aresta. / Graph property testing is the study of randomized sublinear algorithms which decide if an input graph $G$ with $n$ vertices satisfies a given property or if it is necessary to add or remove more than $\\epsilon{n \\choose 2}$ edges to make $G$ satisfy it, for some fixed error parameter $\\epsilon$ . A graph property $P$ is called testable if, for every $\\epsilon > 0$, there is such an algorithm for $P$ whose run time is independent of $n$. One of the most important results in this area is due to Alon and Shapira, who showed that every hereditary graph property is testable. In this work, we show analogous results for tournaments --- complete graphs in which every edge is given an orientation.

Spanning tree congestion was defined by Ostrovskii (2004) as a measure of how well a network can perform if only minimal connection can be maintained. We compute the parameter for several families of graphs. In particular, by partitioning a hypercube into pieces with almost optimal edge-boundaries, we give tight estimates of the parameter thereby disproving a conjecture of Hruska (2008). For a typical random graph, the parameter exhibits a zigzag behaviour reflecting the feature that it is not monotone in the number of edges. This motivates the study of the most congested graphs where we show that any graph is close to a graph with small congestion. Next, we enumerate independent sets. Using the independent set polynomial, we compute the extrema of averages in trees and graphs. Furthermore, we consider inverse problems among trees and resolve a conjecture of Wagner (2009). A result in a more general setting is also proved which answers a question of Alon, Haber and Krivelevich (2011). After briefly considering polynomial invariants of general graphs, we specialize into trees. Three levels of tree distinguishing power are exhibited. We show that polynomials which do not distinguish rooted trees define typically exponentially large equivalence classes. On the other hand, we prove that the rooted Ising polynomial distinguishes rooted trees and that the Negami polynomial determines the subtree polynomial, strengthening results of Bollobás and Riordan (2000) and Martin, Morin and Wagner (2008). The top level consists of the chromatic symmetric function and it is proved to be a complete invariant for caterpillars.

The fact that clustering is perhaps the most used technique for exploratory data analysis is only a semaphore that underlines its fundamental importance. The general problem statement that broadly describes clustering as the identification and classification of patterns into coherent groups also implicitly indicates it's utility in other tasks such as supervised learning. In the past decade and a half there have been two developments that have altered the landscape of research in clustering: One is improved results by the increased use of graph theoretic techniques such as spectral clustering and the other is the study of clustering with respect to its relevance in semi-supervised learning i.e. using unlabeled data for improving prediction accuracies. In this work an attempt is made to make contributions to both these aspects. Thus our contributions are two-fold: First, we identify some general issues with the spectral clustering framework and while working towards a solution, we introduce a new algorithm which we call "Regularity Clustering" which makes an attempt to harness the power of the Szemeredi Regularity Lemma, a remarkable result from extremal graph theory for the task of clustering. Secondly, we investigate some practical and useful strategies for using clustering unlabeled data in boosting prediction accuracy. For all of these contributions we evaluate our methods against existing ones and also apply these ideas in a number of settings.

Szemerédi's Regularity Lemma is powerful tool in Graph Theory, yielding many applications in areas such as Extremal Graph Theory, Combinatorial Number Theory and Theoretical Computer Science. Strong hypergraph extensions of graph regularity techniques were recently given by Nagle, Rödl, Schacht and Skokan, by W.T. Gowers, and subsequently, by T. Tao. These extensions have yielded quite a few non-trivial applications to Extremal Hypergraph Theory, Combinatorial Number Theory and Theoretical Computer Science. A main drawback to the hypergraph regularity techniques above is that they are highly technical. In this thesis, we consider a less technical version of hypergraph regularity which more directly generalizes Szemeredi's regularity lemma for graphs. The tools we discuss won't yield all applications of their stronger relatives, but yield still several applications in extremal hypergraph theory (for so-called linear or simple hypergraphs), including algorithmic ones. This thesis surveys these lighter regularity techiques, and develops three applications of them.

Ramsey Theorem, in the most simple form, states that if we are given a positive integer l, there exists a minimal integer r(l), called the Ramsey number, such any partition of the edges of K_r(l) into two sets, i.e. a 2-coloring, yields a copy of K_l contained entirely in one of the partitioned sets, i.e. a monochromatic copy of Kl. We prove an extension of Ramsey's Theorem, in the more general form, by replacing complete graphs by multipartite graphs in both senses, as the partitioned set and as the desired monochromatic graph. More formally, given integers l and k, there exists an integer p(m) such that any 2-coloring of the edges of the complete multipartite graph K_p(m);r(k) yields a monochromatic copy of K_m;k . The tools that are used to prove this result are the Szemeredi Regularity Lemma and the Blow Up Lemma. A full proof of the Regularity Lemma is given. The Blow-Up Lemma is merely stated, but other graph embedding results are given. It is also shown that certain embedding conditions on classes of graphs, namely (f , ?) -embeddability, provides a method to bound the order of the multipartite Ramsey numbers on the graphs. This provides a method to prove that a large class of graphs, including trees, graphs of bounded degree, and planar graphs, has a linear bound, in terms of the number of vertices, on the multipartite Ramsey number.

Erdos proved the well-known result that every graph has a spanning, bipartite subgraph such that every vertex has degree at least half of its original degree. Bollobas and Scott conjectured that one can get a slightly weaker result if we require the subgraph to be not only spanning and bipartite, but also balanced. We prove this conjecture for graphs of maximum degree 3. The majority of the paper however, will focus on graph tiling. Graph tiling (or sometimes referred to as graph packing) is where, given a graph H, we find a spanning subgraph of some larger graph G that consists entirely of disjoint copies of H. With the Regularity Lemma and the Blow-up Lemma as our main tools, we prove an asymptotic minimum degree condition for an arbitrary bipartite graph G to be tiled by another arbitrary bipartite graph H. This proves a conjecture of Zhao and also implies an asymptotic version of a result of Kuhn and Osthus for bipartite graphs.

In graph theory, as in many fields of mathematics, one is often interested in finding the maxima or minima of certain functions and identifying the points of optimality. We consider a variety of functions on graphs and hypegraphs and determine the structures that optimize them. A central problem in extremal (hyper)graph theory is that of finding the maximum number of edges in a (hyper)graph that does not contain a specified forbidden substructure. Given an integer n, we consider hypergraphs on n vertices that do not contain a strong simplex, a structure closely related to and containing a simplex. We determine that, for n sufficiently large, the number of edges is maximized by a star. We denote by F(G, r, k) the number of edge r-colorings of a graph G that do not contain a monochromatic clique of size k. Given an integer n, we consider the problem of maximizing this function over all graphs on n vertices. We determine that, for large n, the optimal structures are (k − 1)2-partite Turán graphs when r = 4 and k ∈ {3, 4} are fixed. We call a graph F color-critical if it contains an edge whose deletion reduces the chromatic number of F and denote by F(H) the number of copies of the specified color-critical graph F that a graph H contains. Given integers n and m, we consider the minimum of F(H) over all graphs H on n vertices and m edges. The Turán number of F, denoted ex(n, F), is the largest m for which the minimum of F(H) is zero. We determine that the optimal structures are supergraphs of Tur´an graphs when n is large and ex(n, F) ≤ m ≤ ex(n, F)+cn for some c > 0.