## Electronic Theses and Dissertations, 2004-2019

#### Abstract

Chip-firing games and divisor theory on finite, connected, undirected and unweighted graphs have been studied as analogs of divisor theory on Riemann Surfaces. As part of this theory, a version of the one-dimensional Riemann-Roch theorem was introduced for graphs by Matt Baker in 2007. Properties of algebraic curves that have been studied can be applied to study graphs by means of the divisor theory of graphs. In this research, we investigate the property of a vertex of a graph having the Weierstrass property in analogy to the theory of Weierstrass points on algebraic curves. The weight of the Weierstrass vertices is then calculated in a manner analogous to the algebraic curve case. Although there are many graphs for which all vertices are Weierstrass vertices, there are bounds on the total weight of the Weierstrass vertices as a function of the arithmetic genus.For complete graphs, all of the vertices are Weierstrass when the number of vertices (n) is greater than or equals to \$4\$ and no vertex is Weierstrass for \$n\$ strictly less than 4. We study the complete graphs on 4, 5 and 6 vertices and reveal a pattern in the gap sequence for higher cases of n.Furthermore, we introduce a formula to calculate the Weierstrass weight of a vertex of the complete graph on n vertices. Additionally, we prove that Weierstrass semigroup of complete graphs is 2 - generated. Moreover, we show that there are no graphs of genus 2 and 6 vertices with all the vertices being normal Weierstrass vertices and generalize this result to any graph with genus g.

2018

Spring

Brennan, Joseph

#### Degree

Master of Science (M.S.)

#### College

College of Sciences

Mathematics

#### Degree Program

Mathematical Science

application/pdf

CFE0007397

#### URL

http://purl.fcla.edu/fcla/etd/CFE0007397

English

November 2019

1 year

#### Access Status

Masters Thesis (Open Access)

COinS