A path on n vertices, denoted by Pn, is a simple graph whose vertices can be ordered so that two vertices are adjacent if and only if they are consecutive in the order. A fourth power path, Pn4, is obtained from Pn by adding edges between any two vertices, u and v, whose distance in Pn, denoted by dPn(u,v), is less than or equal to four. The diameter of a graph G, denoted diam(G) is the greatest distance between any two distinct vertices of G. A radio labeling of a graph G is a function f that assigns to each vertex a label from the set {0,1,2,...} such that |f(u)−f(v)| ≥ diam(G)−d(u,v)+1 holds for any two distinct vertices, u and v in G (i.e., u, v ∈ V (G)). The greatest value assigned to a vertex by f is called the span of the radio labeling f, i.e., spanf =max{f(v) : v ∈ V (G)}. The radio number of G, rn(G), is the minimum span of f over all radio labelings f of G. In this paper, we provide a lower bound for the radio number of the fourth power path.

A k-majority tournament is a directed graph that models a k-majority voting scenario, which is realized by 2k - 1 rankings, called linear orderings, of the vertices in the tournament. Every k-majority voting scenario can be modeled by a tournament, but not every tournament is a model for a k-majority voting scenario. In this thesis we show that all acyclic tournaments can be realized as 2-majority tournaments. Further, we develop methods to realize certain quadratic residue tournaments as k-majority tournaments. Thus, each tournament within these classes of tournaments is a model for a k-majority voting scenario. We also explore important structures specifically pertaining to 2- and 3-majority tournaments and introduce the idea of pseudo-3-majority tournaments and inherited 2-majority tournaments.

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.

A methodology for 2D and 3D collision free path planning algorithm in a structured environment is presented. The isolated free convex areas are represented as a nodes in a graph, and a graph traversal strategy that dynamically allocates costs to graph path is used. Modification of the algorithm for small computational time and optimality is discussed. The 3D path planning is done in the three orthogonal two-dimensional projections of a 3D environment. Collision checking to increase the optimality for 3D paths is done in each of the three orthogonal two-dimensional subspaces.

Immersion is a notion of graph inclusion related to the notion of graph minors. While the structural theory of graph minors is extensive, there are still numerous open problems in the structural theory of graph immersions. Kuratowski's theorem claims that the class of graphs that do not contain a subdivision of the graphs K3,3 and K5 is exactly the class of planar graphs. The main goal of this thesis is to describe the structure of the graphs that do not contain an immersion of K3,3. Such graphs can be separated by small edge cuts into small graphs or planar 3-regular graphs. 1

The goal of this master thesis is to design and implement an analytical application for Security Information Service by providing a software prototype. The solution proposes an enhancement of existing graph that allows security analytics to analyse, edit and visualize objects and relations that are saved into a relational database. In the thesis, we walk through the process of development step by step. First, we investigate the current version software and the requirements of the customer. Afterwards, we design the architecture to be easily extendable with new modules and reliable libraries. In the next step, we implement the application, present our solution to the customer and conduct excessive testing. The final step is evaluating our solution by comparing it to the current software solution in use.

Graph theory is fascinating branch of math. Leonhard Euler introduced the concept of Graph Theory in his paper about the seven bridges of Konigsberg published in 1736. In a nutshell, graph theory is the study of pair-wise relationships between objects. Each object is represented using a vertex and in case of a relationship between a pair of vertices, they will be connected using an edge. In this dissertation, graph theory is used to study several important combinatorial optimization problems. In chapter 2, we study the multi-dimensional assignment problem using their underlying hypergraphs. It will be shown how the MAP can be represented by a k-partite graph and how any solution to MAP is associated to a k-clique in the respective k-partite graph. Two heuristics are proposed to solve the MAP and computational studies are performed to compare the performance of the proposed methods with exact solutions. On the heels of chapter 3, a new branch-and-bound method is proposed to solve the problem of finding all k-cliques in a k-partite graph. The new method utilizes bitsets as the data-structure to represent graph data. A new pruning method is introduced in BitCLQ, and CPU instructions are used to improve the performance of the branch-and-bound method. BitCLQ gains up to 300% speed up over existing methods. In chapter 4, two new heuristic to solve decomposable cost MAP's are proposed. The proposed heuristic are based on the partitioning of the underlying graph representing the MAP. In the first heuristic method, MAP's are partitioned into several smaller MAP's with the same dimensiality and smaller cardinality. The solution to the original MAP is constructed incrementally, by using the solutions obtained from each of the smaller MAP's. The second heuristic works in the same fashion. But instead of partitioning the graph along the cardinality, graphs are divided into smaller graphs with the same cardinality but smaller dimensionality. The result of each heuristic is then compared with a well-known existing heuristic. An important problem in graph theory is the maximum clique problem (MCQ). A clique in a graph is defined as a complete subgraph. MCQ problem entails finding the size of the largest clique contained in a graph. General branch-and-bound methods to solve MCQ use graph coloring to find an upper bound on the size of the maximum clique. Recently, a new MAX-SAT based branch-and-bound method for MCQ is proposed that improves the quality of the upper bound obtained from graph coloring. In chapter 5, a branch and bound algorithm is proposed for the maximum clique problem. The proposed method uses bitsets as the data-structure. The result of the computational studies to compare the proposed method with existing methods for MCQ is provided. Chapter 6 contains an application of a graph theory in solving a risk management problem. A new mixed-integer mathematical model to formulate a risk-based network is provided. It will be shown that an optimal solution of the model is a maximal clique in the underlying graph representing the network. The model is then solved using a graph-theoretic approach and the results are compared to CPLEX.

<p>Today’s society depends on complex and technically advanced mechanical systems, often containing a variety of different components. Despite careful development andconstruction, some of these components may eventually fail. To avoid unnecessary damage, for example environmental or financial, there is a need to locate and diagnose these faults as fast as possible. This can be done with a diagnostic system, which should produce an alarm if there is a fault in the mechanical system and, if possible, indicate the reason behind it. </p><p>In model based diagnosis, a mathematical model of a fault free system is used to detect if the monitored system contain any faults. This is done by constructing fault indicators, called fault tests, consisting of equations from different parts of the model. Finding these parts is a time-consuming and demanding task, hence it is preferable if as much as possible of this process can be automated. In this thesis an algorithm that finds all parts of a system that can be used to create these fault tests is presented. To make this analysis feasible, in industrial applications, a simplified version of a system model called a structural model is used. Since the models considered in this thesis are implemented in the mathematical software Simulink, a method for transforming Simulink models into analytical equations and structural models is described. As a way of increasing the diagnostic performance for a model based diagnostic system, information about different faults, called fault models, can be included in the model. However, since the models in this thesis are implemented in Simulink, there is no direct way in which this can be preformed. This thesis describes a solution to this problem. The correctness of the algorithms in this thesis are proved and they have been applied, with supreme results, to aScania truck engine model.</p>

Biological data derived from high-throughput microarrays can be transformed into finite, simple, undirected graphs and analyzed using tools first introduced by the Langston Lab at the University of Tennessee. Transforming raw data can be broken down into three main tasks: data normalization, generation of similarity metrics, and threshold selection. The choice of methods used in each of these steps effect the final outcome of the graph, with respect to size, density, and structure. A number of different algorithms are examined and analyzed to illustrate the magnitude of the effects. Graph-based tools are then used to extract putative gene networks. These tools are loosely based on the concept of clique, which generates clusters optimized for density. Innovative additions to the paraclique algorithm, developed at the Langston Lab, are introduced to generate results that have highest average correlation or highest density. A new suite of algorithms is then presented that exploits the use of a priori gene interactions. Aptly named the anchored analysis toolkit, these algorithms use known interactions as anchor points for generating subgraphs, which are then analyzed for their graph structure. This results in clusters that might have otherwise been lost in noise. A main product of this thesis is a novel collection of algorithms to generate exact solutions to the maximum clique problem for graphs that are too large to fit within core memory. No other algorithms are currently known that produce exact solutions to this problem for extremely large graphs. A combination of in-core and out-of-core techniques is used in conjunction with a distributed-memory programming model. These algorithms take into consideration such pitfalls as external disk I/O and hardware failure and recovery. Finally, a web-based tool is described that provides researchers access the aforementioned algorithms. The Graph Algorithms Pipeline for Pathway Analysis tool, GrAPPA, was previously developed by the Langston Lab and provides the software needed to take raw microarray data as input and preprocess, analyze, and post-process it in a single package. GrAPPA also provides access to high-performance computing resources, via the TeraGrid.

Thesis (M.Sc.)--Memorial University of Newfoundland, 2005. / Bibliography: leaves 78-83.