#### 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.

#### Graduation Date

2018

#### Semester

Spring

#### Advisor

Brennan, Joseph

#### Degree

Master of Science (M.S.)

#### College

College of Sciences

#### Department

Mathematics

#### Degree Program

Mathematical Science

#### Format

application/pdf

#### Identifier

CFE0007397

#### URL

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

#### Language

English

#### Release Date

November 2019

#### Length of Campus-only Access

1 year

#### Access Status

Masters Thesis (Open Access)

#### STARS Citation

De Vas Gunasekara, Ajani Ruwandhika Chulangi, "Weierstrass Vertices and Divisor Theory of Graphs" (2018). *Electronic Theses and Dissertations, 2004-2019*. 6248.

https://stars.library.ucf.edu/etd/6248