In this dissertation we first introduce an extension of the notion of parking functions to cars of different sizes. We prove a product formula for the number of such sequences and provide a refinement using a multi-parameter extension of the Abel--Rothe polynomial. Next, we study the incidence Hopf algebra on the noncrossing partition lattice. We demonstrate a bijection between the terms in the canceled chain decomposition of its antipode and noncrossing hypertrees. Thirdly, we analyze the sum of the 𝑟th powers of the descent set statistic on permutations and how many small prime factors occur in these numbers. These results depend upon the base 𝑝 expansion of both the dimension and the power of these statistics. Finally, we inspect the ƒ-vector of the descent polytope DPv, proving a maximization result using an analogue of the boustrophedon transform.

Graphs provide interesting ways to generate families of lattice polytopes. In particular, one can use matrices encoding the information of a finite graph to define vertices of a polytope. This dissertation initiates the study of the Laplacian simplex, PG, obtained from a finite graph G by taking the convex hull of the columns of the Laplacian matrix for G. The Laplacian simplex is extended through the use of a parallel construction with a finite digraph D to obtain the Laplacian polytope, PD. Basic properties of both families of simplices, PG and PD, are established using techniques from Ehrhart theory. Motivated by a well-known conjecture in the field, our investigation focuses on reflexivity, the integer decomposition property, and unimodality of Ehrhart h*-vectors of these polytopes. A systematic investigation of PG for trees, cycles, and complete graphs is provided, which is enhanced by an investigation of PD for cyclic digraphs. We form intriguing connections with other families of simplices and produce G and D such that the h*-vectors of PG and PD exhibit extremal behavior.

Simplices are the "simplest" examples of polytopes, and yet they exhibit much of the rich and subtle combinatorics and commutative algebra of their more general cousins. In this way they are sufficiently complicated --- insights gained from their study can inform broader research in Ehrhart theory and associated fields. In this dissertation we consider two previously unstudied properties of lattice simplices; one algebraic and one combinatorial. The first is the Poincar\'e series of the associated semigroup algebra, which is substantially more complicated than the Hilbert series of that same algebra. The second is the partial ordering of the elements of the fundamental parallelepiped associated to the simplex. We conclude with a proof-of-concept for using machine learning techniques in algebraic combinatorics. Specifically, we attempt to model the integer decomposition property of a family of lattice simplices using a neural network.

The cutwidth of different graphs is a topic that has been extensively studied. The basis of this paper is the cutwidth of complete n-partite graphs. While looking at the cutwidth of complete n-partite graphs, we strictly consider the linear embedding and cyclic embedding. The relationship between the linear cutwidth and the cyclic cutwidth is discussed and used throughout multiple proofs of different cases for the cyclic cutwidth. All the known cases for the linear and cyclic cutwidth of complete bipartite, complete tripartite, and complete n-partite graphs are highlighted. The main focus of this paper is to expand on the cyclic cutwidth of complete tripartite graphs. Using the relationship of the linear cutwidth and cyclic cutwidth of any graph, we find a lower bound and an upper bound for the cyclic cutwidth of complete tripartite graph K_(r,r,pr) where r is odd and p is a natural number. Throughout this proof there are two cases that develop, p even and p odd. Within each case we have to consider the cuts of multiple regions to find the maximum cut of the cyclic embedding. Once all regions within each case are considered, we discover that the upper and lower bounds are equivalent. This discovery of the cyclic cutwidth of complete tripartite graph K_(r,r,pr) where r is odd and p is a natural number results in getting one step closer to finding the cyclic cutwidth of any complete tripartite graph K_(r,s,t).

A. Ádám conjectured that for any non-acyclic digraph D, there exists an arc whose reversal reduces the total number of cycles in D. In this thesis we characterize and identify structure common to all digraphs for which Ádám's conjecture holds. We investigate quasi-acyclic digraphs and verify that Ádám's conjecture holds for such digraphs. We develop the notions of arc-cycle transversals and reversal sets to classify and quantify this structure. It is known that Ádám's conjecture does not hold for certain infinite families of digraphs. We provide constructions for such counterexamples to Ádám's conjecture. Finally, we address a conjecture of Reid [Rei84] that Ádám's conjecture is true for tournaments that are 3-arc-connected but not 4-arc-connected.

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.

Many properties have been found hidden in Pascal's triangle. In this paper, we will present several known properties in Pascal's triangle as well as the properties that lift to different extensions of the triangle, namely Pascal's pyramid and the trinomial triangle. We will tailor our interest towards Fermat numbers and the hockey stick property. We will also show the importance of the hockey stick properties by using them to prove a property in the trinomial triangle.

A 3-map is a 3-region colorable map. They have been studied by Craft and White in their paper 3-maps. This thesis introduces topological graph theory and then investigates 3-maps in detail, including examples, special types of 3-maps, the use of 3-maps to find the genus of special graphs, and a generalization known as n-maps.

A configuration is a finite set of points in the plane. Two configurations have the same order type if there exists a bijection between them that preserves the orientation of every ordered triple. A configuration A contains a copy of a configuration B if some subset of A has the same order type of B and we denote this by B \\subset A. For a configuration B and a integer N, the extremal number ex(N,B)=max{|A|: B ot \\subset A \\subset [N]^2} is the maximum size of a subset of [N]^2 without a copy of B. We give an upper bound for general and convex cases. A random N-set is a set obtained by randomly choosing N points uniformly and independently in the unit square. A configuration is n-universal if contains all order types in general position of size n. We obtain the threshold for the n-universal property up to a log log factor, that is, we obtain integers N_0 and N_1 with log log N_1=O(log log N_0) such that if N >> N_1 (N << N_0), then a random N-set is n-universal with probability tending to 1 (tending to 0). We also determine a bound for the probability of obtaining a random set without a copy of a fixed configuration. / Uma configuração é um conjunto finito de pontos no plano. Duas configurações possuem o mesmo o-tipo se existe uma bijeção entre elas que preserva a orientação de toda tripla orientada. Uma configuração A contém uma cópia da configuração B se algum subconjunto de A possui o mesmo o-tipo que B e denotamos este fato por B \\subset A. Para uma configuração B e um inteiro N, o número extremal ex(N,B)=max{|A|: B ot \\subset A \\subset [N]^2} é o maior tamanho de um subconjunto de [N]^2 sem uma cópia de B. Neste trabalho, determinamos cotas superiores para o caso geral e para o caso convexo. Um N-conjunto aleatório é um conjunto obtido escolhendo N pontos uniformemente e independentemente ao acaso do quadrado unitário. Uma configuração é n-universal se contém todos os o-tipos de tamanho n. Determinamos o limiar da propriedade de um N-conjunto aleatório ser n-universal a menos de erros da ordem de log log, isto é, determinamos inteiros N_0 e N_1 com log log N_0=O(log log N_1) tais que se N >> N_1 (N << N_0), então o N-conjunto aleatório é n-universal com probabilidade tendendo a 1 (tendendo a 0). Também obtivemos cotas para a probabilidade de um conjunto aleatório não possuir determinado o-tipo.

The scheduling and Control Group at the University of South Australia has been studying the optimal control of trains for many years, and has developed in-cab devices that help drivers stay on time and minimise energy use. In this thesis, we re-examine the optimal control theory for the train control problem. In particular, we study the optimal control around steep sections of track. To calculate an optimal driving strategy we need a realistic model of train performance. In particular, we need to know a coefficient of rolling resistance and a coefficient of aerodynamic drag. In practice, these coefficients are different for every train and difficult to predict. In the thesis, we study the use of mathematical filters to estimate model parameters from observations of actual train performance.