We study reducibility for nowhere-zero flows. A reducibility proof typically consists of showing that some induced subgraphs cannot appear in a minimum counter-example to some conjecture. We derive algebraic proofs of reducibility. We define variables which in some sense count the number of nowhere-zero flows of certain type in a graph and then deduce equalities and inequalities that must hold for all graphs. We then show how to use these algebraic expressions to prove reducibility. In our case, these inequalities and equalities are linear. We can thus use the well developed theory of linear programming to obtain certificates of these proof. We make publicly available computer programs we wrote to generate the algebraic expressions and obtain the certificates.

We study several extremal problems in graphs and hypergraphs. The first one is on list-colouring hypergraphs, which is a generalization of the ordinary colouring of hypergraphs. We discuss two methods for determining the list-chromatic number of hypergraphs. One method uses hypergraph polynomials, which invokes Alon's combinatorial nullstellensatz. This method usually requires computer power to complete the calculations needed for even a modest-sized hypergraph. The other method is elementary, and uses the idea of minimum improper colourings. We apply these methods to various classes of hypergraphs, including the projective planes. We focus on solving the list-colouring problem for Steiner triple systems (STS). It is not hard using either method to determine that Steiner triple systems of orders 7, 9 and 13 are 3-list-chromatic. For systems of order 15, we show that they are 4-list-colourable, but they are also ``almost'' 3-list-colourable. For all Steiner triple systems, we prove a couple of simple upper bounds on their list-chromatic numbers. Also, unlike ordinary colouring where a 3-chromatic STS exists for each admissible order, we prove using probabilistic methods that for every $s$, every STS of high enough order is not $s$-list-colourable. The second problem is on embedding nearly-spanning bounded-degree trees in sparse graphs. We determine sufficient conditions based on expansion properties for a sparse graph to embed every nearly-spanning tree of bounded degree. We then apply this to random graphs, addressing a question of Alon, Krivelevich and Sudakov, and determine a probability $p$ where the random graph $G_{n,p}$ asymptotically almost surely contains every tree of bounded degree. This $p$ is nearly optimal in terms of the maximum degree of the trees that we embed. Finally, we solve a problem that arises from quantum computing, which can be formulated as an extremal question about maximizing the size of a type of acyclic directed graph.

This thesis investigates HU+00E4ggkvist & Hell graphs. These graphs are an extension of the idea of Kneser graphs, and as such share many attributes with them. A variety of original results on many different properties of these graphs are given. We begin with an examination of the transitivity and structural properties of HU+00E4ggkvist & Hell graphs. Capitalizing on the known results for Kneser graphs, the exact values of girth, odd girth, and diameter are derived. We also discuss subgraphs of HU+00E4ggkvist & Hell graphs that are isomorphic to subgraphs of Kneser graphs. We then give some background on graph homomorphisms before giving some explicit homomorphisms of HU+00E4ggkvist & Hell graphs that motivate many of our results. Using the theory of equitable partitions we compute some eigenvalues of these graphs. Moving on to independent sets we give several bounds including the ratio bound, which is computed using the least eigenvalue. A bound for the chromatic number is given using the homomorphism to the Kneser graphs, as well as a recursive bound. We then introduce the concept of fractional chromatic number and again give several bounds. Also included are tables of the computed values of these parameters for some small cases. We conclude with a discussion of the broader implications of our results, and give some interesting open problems.

In recent years, there has been a push by automotive manufacturers to improve the efficiency of the vehicle development process. This can be accomplished by creating a computationally efficient vehicle model that has the capability of predicting the vehicle behavior in many different situations at a fast pace. This thesis presents a procedure to automatically generate the simulation code of vehicle systems rolling over three-dimensional (3-D) roads given a description of the model as input. The governing equations describing the vehicle can be formulated using either a numerical or symbolical formulation approach. A numerical approach will re-construct numerical matrices that describe the system at each time step. Whereas a symbolic approach will generate the governing equations that describe the system for all time. The latter method offers many advantages to obtaining the equations. They only have to be formulated once and can be simplified using symbolic simplification techniques, thus making the simulations more computationally efficient. The road model is automatically generated in the formulation stage based on the single elevation function (3-D mathematical function) that is used to represent the road. Symbolic algorithms are adopted to construct and optimize the non-linear equations that are required to determine the contact point. A Newton-Raphson iterative scheme is constructed around the optimized non-linear equations, so that they can be solved at each time step. The road is represented in tabular form when it can not be defined by a single elevation function. A simulation code structure was developed to incorporate the tire on a 3-D road in a symbolic computer implementation of vehicle systems. It was created so that the tire forces and moments that appear in the generalized force matrix can be evaluated during simulation and not during formulation. They are evaluated systematically by performing a number of procedure calls. A road model is first used to determine the contact point between the tire and the ground. Its location is used to calculate the tire intermediate variables, such as the camber angle, that are required by a tire model to evaluate the tire forces and moments. The structured simulation code was implemented in the DynaFlexPro software package by creating a linear graph representation of the tire and the road. DynaFlexPro was used to analyze a vehicle system on six different road profiles performing different braking and cornering maneuvers. The analyzes were repeated in MSC.ADAMS for validation purposes and good agreement was achieved between the two software packages. The results confirmed that the symbolic computing approach presented in this thesis is more computationally efficient than the purely numerical approach. Thus, the simulation code structure increases the versatility of vehicle models by permitting them to be analyzed on 3-D trajectories while remaining computationally efficient.

This dissertation aims at analyzing complex problems arising in the context of dynamical networks, proteomics, and disease prevention. First, a new graph-based method for proving global stability of synchronization in directed dynamical networks is developed. This method utilizes stability and graph theories to clarify the interplay between individual oscillator dynamics and network topology. Secondly, a graph-theoretical algorithm is proposed to predict Ca2+-binding site in proteins. The new algorithm enables us to identify previously-unknown Ca2+-binding sites, and deepens our understanding towards disease-related Ca2+-binding proteins at a molecular level. Finally, an optimization model and algorithm to solve a disease prevention problem are described at the population level. The new resource allocation model is designed to assist clinical managers to make decisions on identifying at-risk population groups, as well as selecting a screening and treatment strategy for chlamydia and gonorrhea patients under a fixed budget. The resource allocation model and algorithm can have a significant impact on real treatment strategy issues.

In typical statistical parametric mapping (SPM) of fMRI data, the functional data are pre-smoothed using a Gaussian kernel to reduce noise at the cost of losing spatial specificity. Wavelet approaches have been incorporated in such analysis by enabling an efficient representation of the underlying brain activity through spatial transformation of the original, un-smoothed data; a successful framework is the wavelet-based statistical parametric mapping (WSPM) which enables integrated wavelet processing and spatial statistical testing. However, in using the conventional wavelets, the functional data are considered to lie on a regular Euclidean space, which is far from reality, since the underlying signal lies within the complex, non rectangular domain of the cerebral cortex. Thus, using wavelets that function on more complex domains such as a graph holds promise. The aim of the current project has been to integrate a recently developed spectral graph wavelet transform as an advanced transformation for fMRI brain data into the WSPM framework. We introduce the design of suitable weighted and un-weighted graphs which are defined based on the convoluted structure of the cerebral cortex. An optimal design of spatially localized spectral graph wavelet frames suitable for the designed large scale graphs is introduced. We have evaluated the proposed graph approach for fMRI analysis on both simulated as well as real data. The results show a superior performance in detecting fine structured, spatially localized activation maps compared to the use of conventional wavelets, as well as normal SPM. The approach is implemented in an SPM compatible manner, and is included as an extension to the WSPM toolbox for SPM.

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. 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.

The cycle space of a strongly connected graph has a basis consisting of directed circuits. The concept of relevant circuits is introduced as a generalization of the relevant cycles in undirected graphs. A polynomial time algorithm for the computation of a minimum weight directed circuit basis is outlined. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

Optimal control methods are applied to mechanical models in order to predict the control strategies in human arm movements. Optimality criteria are used to determine unique controls for a biomechanical model of the human upper-limb with redundant actuators. The motivation for this thesis is to provide a non-task-specific method of motion prediction as a tool for movement researchers and for controlling human models within virtual prototyping environments. The current strategy is based on determining the muscle activation levels (control signals) necessary to perform a task that optimizes several physical determinants of the model such as muscular and joint stresses, as well as performance timing. Currently, the initial and final location, orientation, and velocity of the hand define the desired task. Several models of the human arm were generated using a graph-theoretical method in order to take advantage of similar system topology through the evolution of arm models. Within this framework, muscles were modelled as non-linear actuator components acting between origin and insertion points on rigid body segments. Activation levels of the muscle actuators are considered the control inputs to the arm model. Optimization of the activation levels is performed via a hybrid genetic algorithm (GA) and a sequential quadratic programming (SQP) technique, which provides a globally optimal solution without sacrificing numerical precision, unlike traditional genetic algorithms. Advantages of the underlying genetic algorithm approach are that it does not require any prior knowledge of what might be a 'good' approximation in order for the method to converge, and it enables several objectives to be included in the evaluation of the fitness function. Results indicate that this approach can predict optimal strategies when compared to benchmark minimum-time maneuvers of a robot manipulator. The formulation and integration of the aforementioned components into a working model and the simulation of reaching and lifting tasks represents the bulk of the thesis. Results are compared to motion data collected in the laboratory from a test subject performing the same tasks. Discrepancies in the results are primarily due to model fidelity. However, more complex models are not evaluated due to the additional computational time required. The theoretical approach provides an excellent foundation, but further work is required to increase the computational efficiency of the numerical implementation before proceeding to more complex models.

We can visualize a graph by producing a geometric representation of the graph in which each node is represented by a single point on the plane, and each edge is represented by a curve that connects its two endpoints. Directed graphs are often used to model hierarchical structures; in order to visualize the hierarchy represented by such a graph, it is desirable that a drawing of the graph reflects this hierarchy. This can be achieved by drawing all the edges in the graph such that they all point in an upwards direction. A graph that has a drawing in which all edges point in an upwards direction and in which no edges cross is known as an upward planar graph. Unfortunately, testing if a graph is upward planar is NP-complete. Parameterized complexity is a technique used to find efficient algorithms for hard problems, and in particular, NP-complete problems. The main idea is that the complexity of an algorithm can be constrained, for the most part, to a parameter that describes some aspect of the problem. If the parameter is fixed, the algorithm will run in polynomial time. In this thesis, we investigate contracting an edge in an upward planar graph that has a specified embedding, and show that we can determine whether or not the resulting embedding is upward planar given the orientation of the clockwise and counterclockwise neighbours of the given edge. Using this result, we then show that under certain conditions, we can join two upward planar graphs at a vertex and obtain a new upward planar graph. These two results expand on work done by Hutton and Lubiw. Finally, we show that a biconnected graph has at most <i>k</i>!8^<i>k</i>-1 planar embeddings, where <i>k</i> is the number of triconnected components. By using an algorithm by Bertolazzi et al. that tests whether a given embedding is upward planar, we obtain a parameterized algorithm, where the parameter is the number of triconnected components, for testing the upward planarity of a biconnected graph. This algorithm runs in <i>O</i>(<i>k</i>!8^<i>k</i><i>n</i>³) time.