In this thesis, we show that several natural questions about automatic sequences can be expressed as logical predicates and then decided mechanically. We extend known results in this area to broader classes of sequences (e.g., paperfolding words), introduce new operations that extend the space of possible queries, and show how to process the results. We begin with the fundamental concepts and problems related to automatic sequences, and the corresponding numeration systems. Building on that foundation, we discuss the general logical framework that formalizes the questions we can mechanically answer. We start with a first-order logical theory, and then extend it with additional predicates and operations. Then we explain a slightly different technique that works on a monadic second- order theory, but show that it is ultimately subsumed by an extension of the first-order theory. Next, we give two applications: critical exponent and paperfolding words. In the critical exponent example, we mechanically construct an automaton that describes a set of rational numbers related to a given automatic sequence. Then we give a polynomial-time algorithm to compute the supremum of this rational set, allowing us to compute the critical exponent and many similar quantities. In the paperfolding example, we extend our mechanical procedure to the paperfolding words, an uncountably infinite collection of infinite words. In the following chapter, we address abelian and additive problems on automatic sequences. We give an example of a natural predicate which is provably inexpressible in our first-order theory, and discuss alternate methods for solving abelian and additive problems on automatic sequences. We close with a chapter of open problems, drawn from the earlier chapters.

In recent years quantum algorithms have become a popular area of mathematical research. Farhi and Gutmann introduced the concept of a quantum walk in 1998. In this thesis we investigate mixing properties of continuous-time quantum walks from a mathematical perspective. We focus on the connections between mixing properties and association schemes. There are three main goals of this thesis. Our primary goal is to develop the algebraic groundwork necessary to systematically study mixing properties of continuous-time quantum walks on regular graphs. Using these tools we achieve two additional goals: we construct new families of graphs that admit uniform mixing, and we prove that other families of graphs never admit uniform mixing. We begin by introducing association schemes and continuous-time quantum walks. Within this framework we develop specific algebraic machinery to tackle the uniform mixing problem. Our main algebraic result shows that if a graph has an irrational eigenvalue, then its transition matrix has at least one transcendental coordinate at all nonzero times. Next we study algebraic varieties related to uniform mixing to determine information about the coordinates of the corresponding transition matrices. Combining this with our main algebraic result we prove that uniform mixing does not occur on even cycles or prime cycles. However, we show that the probability distribution of a quantum walk on a prime cycle gets arbitrarily close to uniform. Finally we consider uniform mixing on Cayley graphs of elementary abelian groups. We utilize graph quotients to connect the mixing properties of these graphs to Hamming graphs. This enables us to find new results about uniform mixing on Cayley graphs of certain elementary abelian groups.

We consider the problem of counting <i>transitive factorizations</i> of permutations; that is, we study tuples (&sigma;<i>r</i>,. . . ,&sigma;1) of permutations on {1,. . . ,<i>n</i>} such that (1) the product &sigma;<i>r</i>. . . &sigma;1 is equal to a given target permutation &pi;, and (2) the group generated by the factors &sigma;<i>i</i> acts transitively on {1,. . . ,<i>n</i>}. This problem is widely known as the <i>Hurwitz Enumeration Problem</i>, since an encoding due to Hurwitz shows it to be equivalent to the enumeration of connected branched coverings of the sphere by a surface of given genus with specified branching. Much of our work concerns the enumeration of transitive factorizations of permutations into a minimal number of transposition factors. This problem has received considerable attention, and a formula for the number <i>c</i>(&pi;) of such factorizations of an arbitrary permutation &pi; has been derived through various means. The formula is remarkably simple, being a product of well-known combinatorial numbers, but no bijective proof of it is known except in the special case where &pi; is a full cycle. A major goal of this thesis is to provide further combinatorial rationale for this formula. We begin by introducing an encoding of factorizations (into transpositions) as edge-labelled maps. Our central result is a bijection that allows trees to be "pruned" from such maps. This is shown to explain the appearance of factors of the form <i>k^k</i> in the aforementioned formula for <i>c</i>(&pi;). It also has the effect of shifting focus to the combinatorics of smooth maps (<i>i. e. </i> maps without vertices of degree one). By providing decompositions for certain smooth planar maps, we are able to give combinatorial evaluations of <i>c</i>(&pi;) when &pi; is composed of up to three cycles. Many of these results are generalized to factorizations in which the factors are cycles of any length. We also investigate the <i>Double Hurwitz Problem</i>, which calls for the enumeration of factorizations whose leftmost factor is of specified cycle type, and whose remaining factors are transpositions. Finally, we extend our methods to the enumeration of factorizations up to an equivalence relation induced by possible commutations between adjacent factors.

The study of self-duality has attracted some attention over the past decade. A good deal of research in that time has been done on constructing and classifying all self-dual graphs and in particular polyhedra. We will give an overview of the recent research in the first two chapters. In the third chapter, we will show the necessary condition that a self-complementary self-dual graph have <i>n</i> &#8801; 0, 1 (mod 8) vertices and we will review White's infinite class (the Paley graphs, for which <i>n</i> &#8801; 1 (mod 8)). Finally, we will construct a new infinite class of self-complementary self-dual graphs for which <i>n</i> &#8801; 0 (mod 8).

This thesis investigates dominating sets in Kneser graphs as well as a selection of other topics related to graph domination. Dominating sets in Kneser graphs, especially those of minimum size, often correspond to interesting combinatorial incidence structures. We begin with background on the dominating set problem and a review of known bounds, focusing on algebraic bounds. We then consider this problem in the Kneser graphs, discussing basic results and previous work. We compute the domination number for a few of the Kneser graphs with the aid of some original results. We also investigate the connections between Kneser graph domination and the theory of combinatorial designs, and introduce a new type of design that generalizes the properties of dominating sets in Kneser graphs. We then consider dominating sets in the vector space analogue of Kneser graphs. We end by highlighting connections between the dominating set problem and other areas of combinatorics. Conjectures and open problems abound.

The facility location problem is the task of optimally placing a given number of facilities in a certain subset of the plane. In this thesis, we present various mathematical programming formulations of the planar facility location problem, where potential facility locations are not specified. We first consider mixed-integer programming formulations of the planar facility locations problems with squared Euclidean and rectangular distance metrics to solve this problem to provable optimality. We also investigate a heuristic approach to solving the problem by extending the $K$-means clustering algorithm and formulating the facility location problem as a variant of a semidefinite programming problem, leading to a relaxation algorithm. We present computational results for the mixed-integer formulations, as well as compare the objective values resulting from the relaxation algorithm and the modified $K$-means heuristic. In addition, we briefly discuss some of the practical issues related to the facility location model under the continuous customer distribution.

In this thesis, we study the behaviour of Lovasz and Schrijver's lift-and-project operators N and N_0 while being applied recursively to the fractional stable set polytope of a graph. We focus on two related conjectures proposed by Liptak and Tuncel: the N-N_0 Conjecture and Rank Conjecture. First, we look at the algebraic derivation of new valid inequalities by the operators N and N_0. We then present algebraic characterizations of these valid inequalities. Tightly based on our algebraic characterizations, we give an alternate proof of a result of Lovasz and Schrijver, establishing the equivalence of N and N_0 operators on the fractional stable set polytope. Since the above mentioned conjectures involve also the recursive applications of N and N_0 operators, we also study the valid inequalities obtained by these lift-and-project operators after two applications. We show that the N-N_0 Conjecture is false, while the Rank Conjecture is true for all graphs with no more than 8 nodes.

Rayleigh matroids are a class of matroids with sets of bases that satisfy a strong negative correlation property. Interesting characteristics include the existence of an efficient algorithm for sampling the bases of a Rayleigh matroid [7]. It has been conjectured that the class of Rayleigh matroids satisfies Mason’s conjecture [14]. Though many elementary properties of Rayleigh matroids have been established, it is not known if this class has a finite set of minimal excluded minors. At this time, it seems unlikely that this is the case. It has been shown that there is a single minimal excluded minor for the smaller class of binary Rayleigh matroids [5]. The aim of this thesis is to detail our search for the set of minimal excluded minors for ternary Rayleigh matroids. We have found several minimal excluded minors for the above class of matroids. However, our search is incomplete. It is unclear whether the set of excluded minors for this set of matroids is finite or not, and, if finite, what the complete set of minimal excluded minors is. For our method to answer this question definitively will require a new computer program. This program would automate a step in our process that we have done by hand: writing polynomials in at least ten indeterminates as a sum with many terms, squared.

This thesis is devoted to the analysis of a class of iterative probabilistic algorithms in regular graphs, called locally greedy algorithms, which will provide bounds for graph functions in regular graphs with large girth. This class is useful because, by conveniently setting the parameters associated with it, we may derive algorithms for some well-known graph problems, such as algorithms to find a large independent set, a large induced forest, or even a small dominating set in an input graph G. The name ``locally greedy" comes from the fact that, in an algorithm of this class, the probability associated with the random selection of a vertex v is determined by the current state of the vertices within some fixed distance of v. Given r > 2 and an r-regular graph G, we determine the expected performance of a locally greedy algorithm in G, depending on the girth g of the input and on the degree r of its vertices. When the girth of the graph is sufficiently large, this analysis leads to new lower bounds on the independence number of G and on the maximum number of vertices in an induced forest in G, which, in both cases, improve the bounds previously known. It also implies bounds on the same functions in graphs with large girth and maximum degree r and in random regular graphs. As a matter of fact, the asymptotic lower bounds on the cardinality of a maximum induced forest in a random regular graph improve earlier bounds, while, for independent sets, our bounds coincide with asymptotic lower bounds first obtained by Wormald. Our result provides an alternative proof of these bounds which avoids sharp concentration arguments. The main contribution of this work lies in the method presented rather than in these particular new bounds. This method allows us, in some sense, to directly analyse prioritised algorithms in regular graphs, so that the class of locally greedy algorithms, or slight modifications thereof, may be applied to a wider range of problems in regular graphs with large girth.

This thesis focuses on convex sets and convex cones defined using hyperbolic polynomials. We first review some of the theory of convex sets in $\R^d$ in general. We then review some classical algebraic theorems concerning polynomials in a single variable, as well as presenting a few more modern results about them. We then discuss the theory of hyperbolic polynomials in several variables and their associated hyperbolicity cones. We survey various ways to build and decompose hyperbolic cones and we prove that every nontrivial hyperbolic cone is the intersection of its derivative cones. We conclude with a brief discussion of the set of extreme rays of a hyperbolic cone.