If R is a list of positive integers with greatest common denominator equal to 1, calculating the Frobenius number of R is in general NP-hard. Dino Lorenzini defines the arithmetical graph, which naturally arises in arithmetic geometry, and a notion of genus, the g-number, that in specific cases coincides with the Frobenius number of R. A result of Dino Lorenzini's gives a method for quickly calculating upper bounds for the g-number of arithmetical graphs. We discuss the arithmetic geometry related to arithmetical graphs and present an example of an arithmetical graph that arises in this context. We also discuss the construction for Lorenzini's Riemann-Roch structure and how it relates to the Riemann-Roch theorem for finite graphs shown by Matthew Baker and Serguei Norine. We then focus on the connection between the Frobenius number and arithmetical graphs. Using the Laplacian of an arithmetical graph and a formulation of chip-firing on the vertices of an arithmetical graph, we show results that can be used to find arithmetical graphs whose g-numbers correspond to the Frobenius number of R. We describe how this can be used to quickly calculate upper bounds for the Frobenius number of R.

To determine the function of a protein, we must know its 3-dimensional structure, which can be difficult to ascertain. Currently, predictive models are used to determine the structure of a protein from its sequence, but these models do not always predict the correct structure. To this end we use a nested graph model along with weighted invariants to minimize the errors and improve the accuracy of a predictive model to determine if we have the correct structure for a protein.

Let $M_v$ denotes a complete mixed graph on $v$ vertices, and let $S_6^i$ denotes the partial orientation of the 6-star with twice as many arcs as edges. In this work, we state and prove the necessary and sufficient conditions for the existence of $\lambda$-fold decomposition of a complete mixed graph into $S_6^i$ for $i\in\{1,2,3,4\}$. We used the difference method for our proof in some cases. We also give some general sufficient conditions for the existence of $S_6^i$-decomposition of the complete bipartite mixed graph for $i\in\{1,2,3,4\}$. Finally, this work introduces the decomposition of a complete mixed graph with a hole into mixed stars.

The arcsine distribution looks at the fraction of time one player is winning in a fair coin toss game and has been studied for over a hundred years. There has been little further work on how the distribution changes when the coin tosses are not fair or when a player has already won the initial coin tosses or, equivalently, starts with a lead. This thesis will first cover a proof of the arcsine distribution. Then, we explore how the distribution changes when the coin the is unfair. Finally, we will explore the distribution when one person has won the first few flips.

A set of elements with a binary operation is called a system, or, more explicitly, a mathematical system. The following discussion will involve systems with only one operation. This operation will be denoted by "⋅" and will sometimes be referred to as a product. A system, S, of n elements (x1, x2, ..., xn) is associative if xi ⋅ (xj ⋅ xk) = (xi ⋅ xj) ⋅ xk for all i, j, k ≤ n. In a modern algebra class the following problem was proposed. What is the least number of elements a system can have and be non-associative? A system, S, of n elements (x1, x2, ..., xn) is associative if xi ⋅ (xj ⋅ xk) /= (xi ⋅ xj) ⋅ xk for some i, j, k ≤ n. It is obvious that a system of one element must be associative. Any binary operation could have but one result. A nonassociative system of two elements (a, b) can be constructed by letting a ⋅ a = b⋅a = b. , a⋅(a⋅a) = a⋅b and (a⋅a)⋅a = b⋅a = b. If a⋅b = a, then a⋅(a⋅a) /= (a⋅a)⋅a Thus the system is nonassociative. As is often the case this question leads to others. Are there systems of n elements such that xi ⋅ (xj ⋅ xk) /= (xi ⋅ xj) ⋅ xk for all i, j, k ≤ n? If such systems exist, what are their charcateristics? Such questions as these led to the development of this paper. A system, S, of n elements such that xi ⋅ (xj ⋅ xk) /= (xi ⋅ xj) ⋅ xk for all i, j, k ≤ n is called an anti-associative system. The purpose of this paper is to establish the existence of antiassociative systems of n elements and to find characteristics of these systems in as much detail as possible. Propositions will first be considered that apply to anti-associative systems in general. Then anti-associative systems of two, three, and four elements will be obtained. The general results that each of these special cases lead to will be developed. A special type of anti-associative system will be considered. These special anti-associative systems suggest a broader field. For a set of elements a group of classes of systems is defined. The operation may associative, anti-associative, or neither. Many questions are let unanswered as to the characteristics of anti-associative systems, but this paper opens new avenues to attack a broader problem.

In a paper by Beeler and Hoilman, the traditional game of peg solitaire is generalized to graphs in the combinatorial sense. One of the important open problems in this paper was to classify solvable trees. In this thesis, we will give necessary and sufficient conditions for the solvability for all trees with diameter four. We also give the maximum number of pegs that can be left on such a graph under the restriction that we jump whenever possible.

A minimal covering of a graph G with isomorphic copies of graph H is a set {H1, H2, H3, ... , Hn} where Hi is isomorphic to H, the vertex set of Hi is a subset of G, the edge set of G is a subset of the union of Hi's, and the cardinality of the union of Hi's minus G is minimum. Some studies have been made of covering the complete graph in which case an added condition of the edge set of Hi is the subset of the edge set of G for all i which implies no additional restrictions. However, if G is not the complete graph, then this condition may have implications. We will give necessary and sufficient conditions for minimal coverings of complete bipartite graph with 6-cycles, which we call minimal unrestricted coverings. We also give necessary and sufficient conditions for minimal coverings of the complete bipartite graph with 6-cycles with the added condition the edge set of Hi is a subset of G for all i, and call these minimal restricted coverings.

In this thesis, we consider packings and coverings of various complete graphs with the 4-cycle with a pendant edge. We consider both restricted and unrestricted coverings. Necessary and sufficient conditions are given for such structures for (1) complete graphs Kv, (2) complete bipartite graphs Km,n, and (3) complete graphs with a hole K(v,w).

In this thesis, we study universal hypergraphs. What are these? Let us start with defining a universal graph as a graph on n vertices that contains each of the many possible graphs of a smaller size k < n as an induced subgraph. A hypergraph is a discrete structure on n vertices in which edges can be of any size, unlike graphs, where the edge size is always two. If all edges are of size three, then the hypergraph is said to be 3-uniform. If a 3-uniform hypergraph can have edges colored one of a colors, then it is called a 3-uniform hypergraph with a colors. Analogously with universal graphs, a universal, induced, 3-uniform, k-hypergraph, with a possible edge colors is then defined to be a 3-uniform a-colored hypergraph on n vertices that contains each of the many possible 3-uniform a-colored hypergraphs on k vertices, k < n. In this thesis, we study conditions for the existence of a such a universal hypergraph, and address the question of how large n must be, given a fixed k, so that hypergraphs on n vertices are universal with high probability. This extends the work of Alon, [2] who studied the case of a = 2, and that too for graphs (not hypergraphs).

In this thesis we will study conditions for the existence of minimal sized omnipatterns in higher dimensions. We will introduce recent work conducted on one dimensional and two dimensional patterns known as omnisequences and omnimosaics, respectively. These have been studied by Abraham et al [3] and Banks et al [2]. The three dimensional patterns we study are called omnisculptures, and will be the focus of this thesis. A (K,a) omnisequence of length n is a string of letters that contains each of the ak words of length k over [A]={1,2,...a} as a substring. An omnimosaic O(n,k,a) is an n × n matrix, with entries from the set A ={1,2,...,a}, that contains each of the {ak2} k × k matrices over A as a submatrix. An omnisculpture is an n × n × n sculpture (a three dimensional matrix) with entries from set A ={1,2,...,a} that contains all the ak3 k × k × k subsculptures as an embedded submatrix of the larger sculpture. We will show that for given k, the existence of a minimal omnisculpture is guaranteed when kak2/3/e ≤ n ≤kak2/3/e(1+ε) and ε=εk → 0 is a sufficiently small function of k.