This thesis will study a variety of problems in graph theory. Initially, the focus will be on finding minimal degree conditions which guarantee the existence of various subgraphs. These subgraphs will all be formed of cycles, and this area of work will fall broadly into two main categories. First to be considered are cycles in edge-coloured graphs and, in particular, two questions of Li, Nikiforov and Schelp. It will be shown that a 2-edge-coloured graph with minimal degree at least 3n/4 either is isomorphic to the complete 4-partite graph with classes of order n/4, or contains monochromatic cycles of all lengths between 4 and n/2 (rounded up). This answers a conjecture of Li, Nikiforov and Schelp. Attention will then turn to the length of the longest monochromatic cycle in a 2-edge-coloured graph with minimal degree at least cn. In particular, a lower bound for this quantity will be proved which is asymptotically best possible. The next chapter of the thesis then shows that a hamiltonian graph with minimal degree at least (5-sqrt7)n/6 contains a 2-factor with two components. The thesis then concludes with a chapter about X_H, which is the number of copies of a graph H in the random graph G(n,p). In particular, it will be shown that, for a connected graph H, the value of X_H modulo k is approximately uniformly distributed, provided that k is not too large a function of n.

In this thesis we consider two sets of combinatorial structures defined on an Eulerian graph: the Eulerian orientations and Euler tours. We are interested in the computational problems of counting (computing the number of elements in the set) and sampling (generating a random element of the set). Specifically, we are interested in the question of when there exists an efficient algorithm for counting or sampling the elements of either set. The Eulerian orientations of a number of classes of planar lattices are of practical significance as they correspond to configurations of certain models studied in statistical physics. In 1992 Mihail and Winkler showed that counting Eulerian orientations of a general Eulerian graph is #P-complete and demonstrated that the problem of sampling an Eulerian orientation can be reduced to the tractable problem of sampling a perfect matching of a bipartite graph. We present a proof that this problem remains #Pcomplete when the input is restricted to being a planar graph, and analyse a natural algorithm for generating random Eulerian orientations of one of the afore-mentioned planar lattices. Moreover, we make some progress towards classifying the range of planar graphs on which this algorithm is rapidly mixing by exhibiting an infinite class of planar graphs for which the algorithm will always take an exponential amount of time to converge. The problem of counting the Euler tours of undirected graphs has proven to be less amenable to analysis than that of Eulerian orientations. Although it has been known for many years that the number of Euler tours of any directed graph can be computed in polynomial time, until recently very little was known about the complexity of counting Euler tours of an undirected graph. Brightwell and Winkler showed that this problem is #P-complete in 2005 and, apart from a few very simple examples, e.g., series-parellel graphs, there are no known tractable cases, nor are there any good reasons to believe the problem to be intractable. Moreover, despite several unsuccessful attempts, there has been no progress made on the question of approximability. Indeed, this problem was considered to be one of the more difficult open problems in approximate counting since long before the complexity of exact counting was resolved. By considering a randomised input model, we are able to show that a very simple algorithm can sample or approximately count the Euler tours of almost every d-in/d-out directed graph in expected polynomial time. Then, we present some partial results towards showing that this algorithm can be used to sample or approximately count the Euler tours of almost every 2d-regular graph in expected polynomial time. We also provide some empirical evidence to support the unproven conjecture required to obtain this result. As a sideresult of this work, we obtain an asymptotic characterisation of the distribution of the number of Eulerian orientations of a random 2d-regular graph.

Neste trabalho, aplicamos o algoritmo Breadth-First Search para encontrar o tamanho de uma componente conectada no grafo aleatório de Erdos-Rényi. Uma cadeia de Markov é obtida deste procedimento. Apresentamos alguns resultados bem conhecidos sobre o comportamento dessa cadeia de Markov. Combinamos alguns destes resultados para obter uma proposição sobre a probabilidade da componente atingir um determinado tamanho e um resultado de convergência do estado da cadeia neste instante. Posteriormente, aplicamos o teorema de convergência de Darling (2002) a sequência de cadeias de Markov reescaladas e indexadas por N, o número de vértices do grafo, para mostrar que as trajetórias dessas cadeias convergem uniformemente em probabilidade para a solução de uma equação diferencial ordinária. Deste resultado segue a bem conhecida lei fraca dos grandes números para a componente gigante do grafo aleatório de Erdos-Rényi, no caso supercrítico. Além disso, obtemos o limite do fluído para um modelo epidêmico que é uma extensão daquele proposto em Kurtz et al. (2008). / In this work, we apply the Breadth-First Search algorithm to find the size of a connected component of the Erdos-Rényi random graph. A Markov chain is obtained of this procedure. We present some well-known results about the behavior of this Markov chain, and combine some of these results to obtain a proposition about the probability that the component reaches a certain size and a convergence result about the state of the chain at that time. Next, we apply the convergence theorem of Darling (2002) to the sequence of rescaled Markov chains indexed by N, the number of vertices of the graph, to show that the trajectories of these chains converge uniformly in probability to the solution of an ordinary dierential equation. From the latter result follows the well-known weak law of large numbers of the giant component of the Erdos-Renyi random graph, in the supercritical case. Moreover, we obtain the uid limit for an epidemic model which is an extension of that proposed in Kurtz et al. (2008).

The k-core of a graph is its maximal subgraph with minimum degree at least k. The study of k-cores in random graphs was initiated by Bollobás in 1984 in connection to k-connected subgraphs of random graphs. Subsequently, k-cores and their properties have been extensively investigated in random graphs and hypergraphs, with the determination of the threshold for the emergence of a giant k-core, due to Pittel, Spencer and Wormald, as one of the most prominent results. In this thesis, we obtain an asymptotic formula for the number of 2-connected graphs, as well as 2-edge-connected graphs, with given number of vertices and edges in the sparse range by exploiting properties of random 2-cores. Our results essentially cover the whole range for which asymptotic formulae were not described before. This is joint work with G. Kemkes and N. Wormald. By defining and analysing a core-type structure for uniform hypergraphs, we obtain an asymptotic formula for the number of connected 3-uniform hypergraphs with given number of vertices and edges in a sparse range. This is joint work with N. Wormald. We also examine robustness aspects of k-cores of random graphs. More specifically, we investigate the effect that the deletion of a random edge has in the k-core as follows: we delete a random edge from the k-core, obtain the k-core of the resulting graph, and compare its order with the original k-core. For this investigation we obtain results for the giant k-core for Erdős-Rényi random graphs as well as for random graphs with minimum degree at least k and given number of vertices and edges.

Statistical analysis of network data is an active field of study, in which researchers inves- tigate graph-theoretic concepts and various probability models that explain the behaviour of real networks. This thesis attempts to combine two of these concepts: an exponential random graph and a centrality index. Exponential random graphs comprise the most useful class of probability models for network data. These models often require the assumption of a complex dependence structure, which creates certain difficulties in the estimation of unknown model parameters. However, in the context of dynamic networks the exponential random graph model provides the opportunity to incorporate a complex network structure such as centrality without the usual drawbacks associated with parameter estimation. The thesis employs this idea by proposing probability models that are equivalent to the logistic regression models and that can be used to explain behaviour of both static and dynamic networks.

The k-core of a graph is its maximal subgraph with minimum degree at least k. The study of k-cores in random graphs was initiated by Bollobás in 1984 in connection to k-connected subgraphs of random graphs. Subsequently, k-cores and their properties have been extensively investigated in random graphs and hypergraphs, with the determination of the threshold for the emergence of a giant k-core, due to Pittel, Spencer and Wormald, as one of the most prominent results. In this thesis, we obtain an asymptotic formula for the number of 2-connected graphs, as well as 2-edge-connected graphs, with given number of vertices and edges in the sparse range by exploiting properties of random 2-cores. Our results essentially cover the whole range for which asymptotic formulae were not described before. This is joint work with G. Kemkes and N. Wormald. By defining and analysing a core-type structure for uniform hypergraphs, we obtain an asymptotic formula for the number of connected 3-uniform hypergraphs with given number of vertices and edges in a sparse range. This is joint work with N. Wormald. We also examine robustness aspects of k-cores of random graphs. More specifically, we investigate the effect that the deletion of a random edge has in the k-core as follows: we delete a random edge from the k-core, obtain the k-core of the resulting graph, and compare its order with the original k-core. For this investigation we obtain results for the giant k-core for Erdős-Rényi random graphs as well as for random graphs with minimum degree at least k and given number of vertices and edges.

Statistical analysis of network data is an active field of study, in which researchers inves- tigate graph-theoretic concepts and various probability models that explain the behaviour of real networks. This thesis attempts to combine two of these concepts: an exponential random graph and a centrality index. Exponential random graphs comprise the most useful class of probability models for network data. These models often require the assumption of a complex dependence structure, which creates certain difficulties in the estimation of unknown model parameters. However, in the context of dynamic networks the exponential random graph model provides the opportunity to incorporate a complex network structure such as centrality without the usual drawbacks associated with parameter estimation. The thesis employs this idea by proposing probability models that are equivalent to the logistic regression models and that can be used to explain behaviour of both static and dynamic networks.

This paper is a compilation of results in combinatorics utilizing the probabilistic method. Below is a brief description of the results highlighted in each chapter. Chapter 1 provides basic definitions, lemmas, and theorems from graph theory, asymptotic analysis, and probability which will be used throughout the paper. Chapter 2 introduces the independent domination number. It is then shown that in the random graph model G(n,p) with probability tending to one, the independent domination number is one of two values. Also, the the number of independent dominating sets of given cardinality is analyzed statistically. Chapter 3 introduces the tree domination number. It is then shown that in the random graph model G(n,p) with probability tending to one, the tree domination number is one of two values. Additional related domination parameters are also discussed. Chapter 4 introduces a generalized rook polynomial first studied by J. Goldman et al. Central and local limit theorems are then proven for certain classes of the generalized rook polynomial. Special cases include known central and local limit theorems for the Stirling numbers of the first and second kind and additionally new limit theorems for the Lah numbers and certain classes of known generalized Stirling numbers. Chapter 5 introduces the Kneser Graph. The exact expected value and variance of the distance between [n] and a vertex chosen uniformly at random is given. An asymptotic formula for the expectation is found.

abstract: The purpose of information source detection problem (or called rumor source detection) is to identify the source of information diffusion in networks based on available observations like the states of the nodes and the timestamps at which nodes adopted the information (or called infected). The solution of the problem can be used to answer a wide range of important questions in epidemiology, computer network security, etc. This dissertation studies the fundamental theory and the design of efficient and robust algorithms for the information source detection problem. For tree networks, the maximum a posterior (MAP) estimator of the information source is derived under the independent cascades (IC) model with a complete snapshot and a Short-Fat Tree (SFT) algorithm is proposed for general networks based on the MAP estimator. Furthermore, the following possibility and impossibility results are established on the Erdos-Renyi (ER) random graph: $(i)$ when the infection duration $<\frac{2}{3}t_u,$ SFT identifies the source with probability one asymptotically, where $t_u=\left\lceil\frac{\log n}{\log \mu}\right\rceil+2$ and $\mu$ is the average node degree, $(ii)$ when the infection duration $>t_u,$ the probability of identifying the source approaches zero asymptotically under any algorithm; and $(iii)$ when infection duration $<t_u,$ the breadth-first search (BFS) tree starting from the source is a fat tree. Numerical experiments on tree networks, the ER random graphs and real world networks show that the SFT algorithm outperforms existing algorithms. In practice, other than the nodes' states, side information like partial timestamps may also be available. Such information provides important insights of the diffusion process. To utilize the partial timestamps, the information source detection problem is formulated as a ranking problem on graphs and two ranking algorithms, cost-based ranking (CR) and tree-based ranking (TR), are proposed. Extensive experimental evaluations of synthetic data of different diffusion models and real world data demonstrate the effectiveness and robustness of CR and TR compared with existing algorithms. / Dissertation/Thesis / Doctoral Dissertation Electrical Engineering 2015

Neste trabalho, aplicamos o algoritmo Breadth-First Search para encontrar o tamanho de uma componente conectada no grafo aleatório de Erdos-Rényi. Uma cadeia de Markov é obtida deste procedimento. Apresentamos alguns resultados bem conhecidos sobre o comportamento dessa cadeia de Markov. Combinamos alguns destes resultados para obter uma proposição sobre a probabilidade da componente atingir um determinado tamanho e um resultado de convergência do estado da cadeia neste instante. Posteriormente, aplicamos o teorema de convergência de Darling (2002) a sequência de cadeias de Markov reescaladas e indexadas por N, o número de vértices do grafo, para mostrar que as trajetórias dessas cadeias convergem uniformemente em probabilidade para a solução de uma equação diferencial ordinária. Deste resultado segue a bem conhecida lei fraca dos grandes números para a componente gigante do grafo aleatório de Erdos-Rényi, no caso supercrítico. Além disso, obtemos o limite do fluído para um modelo epidêmico que é uma extensão daquele proposto em Kurtz et al. (2008). / In this work, we apply the Breadth-First Search algorithm to find the size of a connected component of the Erdos-Rényi random graph. A Markov chain is obtained of this procedure. We present some well-known results about the behavior of this Markov chain, and combine some of these results to obtain a proposition about the probability that the component reaches a certain size and a convergence result about the state of the chain at that time. Next, we apply the convergence theorem of Darling (2002) to the sequence of rescaled Markov chains indexed by N, the number of vertices of the graph, to show that the trajectories of these chains converge uniformly in probability to the solution of an ordinary dierential equation. From the latter result follows the well-known weak law of large numbers of the giant component of the Erdos-Renyi random graph, in the supercritical case. Moreover, we obtain the uid limit for an epidemic model which is an extension of that proposed in Kurtz et al. (2008).