Discrete Mathematics and Combinatorics Commons^™

Surjectivity Of The Wahl Map On Cubic Graphs, Angela C. Hanson

Theses and Dissertations--Mathematics

Much of algebraic geometry is the study of curves. One tool we use to study curves is whether they can be embedded in a K3 surface or not. If the Wahl map is surjective on a curve, that curve cannot be embedded in a K3 surface. Therefore, studying if the Wahl map is surjective for a particular curve gives us more insight into the properties of that curve. We simplify this problem by converting graph curves to dual graphs. Then the information for graphs can be used to study the underlying curves. We will discuss conditions for the Wahl map …

Methods Of Computing Graph Gonalities, Noah Speeter

Theses and Dissertations--Mathematics

Chip firing is a category of games played on graphs. The gonality of a graph tells us how many chips are needed to win one variation of the chip firing game. The focus of this dissertation is to provide a variety of new strategies to compute the gonality of various graph families. One family of graphs which this dissertation is particularly interested in is rook graphs. Rook graphs are the Cartesian product of two or more complete graphs and we prove that the gonality of two dimensional rook graphs is the expected value of (n − 1)m where n is …

Geometric And Combinatorial Properties Of Lattice Polytopes Defined From Graphs, Kaitlin Bruegge

Theses and Dissertations--Mathematics

Polytopes are geometric objects that generalize polygons in the plane and polyhedra in 3-dimensional space. Of particular interest in geometric combinatorics are families of lattice polytopes defined from combinatorial objects, such as graphs. In particular, this dissertation studies symmetric edge polytopes (SEPs), defined from simple undirected graphs. In 2019, Higashitani, Jochemko, and Michalek gave a combinatorial description of the hyperplanes that support facets of a symmetric edge polytope in terms of certain labelings of the underlying graph.
Using this framework, we explore the number of facets that can be attained by the symmetric edge polytopes for graphs with certain structure. …

Q-Polymatroids And Their Application To Rank-Metric Codes., Benjamin Jany

Theses and Dissertations--Mathematics

Matroid theory was first introduced to generalize the notion of linear independence. Since its introduction, the theory has found many applications in various areas of mathematics including coding theory. In recent years, q-matroids, the q-analogue of matroids, were reintroduced and found to be closely related to the theory of linear vector rank metric codes. This relation was then generalized to q-polymatroids and linear matrix rank metric codes. This dissertation aims at developing the theory of q-(poly)matroid and its relation to the theory of rank metric codes. In a first part, we recall and establish preliminary results for both q-polymatroids and …

Geometry Of Pipe Dream Complexes, Benjamin Reese

Theses and Dissertations--Mathematics

In this dissertation we study the geometry of pipe dream complexes with the goal of gaining a deeper understanding of Schubert polynomials. Given a pipe dream complex PD(w) for w a permutation in the symmetric group, we show its boundary is Whitney stratified by the set of all pipe dream complexes PD(v) where v > w in the strong Bruhat order. For permutations w in the symmetric group on n elements, we introduce the pipe dream complex poset P(n). The dual of this graded poset naturally corresponds to the poset of strata associated to the Whitney stratification of the boundary of …

Lattice Minors And Eulerian Posets, William Gustafson

Theses and Dissertations--Mathematics

We study a partial ordering on pairings called the uncrossing poset, which first appeared in the literature in connection with a certain stratified space of planar electrical networks. We begin by examining some of the relationships between the uncrossing poset and Catalan combinatorics, and then proceed to study the structure of lower intervals. We characterize the lower intervals in the uncrossing poset that are isomorphic to the face lattice of a cube. Moving up in complexity certain lower intervals are isomorphic to the poset of simple vertex labeled minors of an associated graph.

Inspired by this structure, we define a …