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Weierstrass vertices and divisor theory of graphs

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Date Issued:
2018
Abstract/Description:
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.
Title: Weierstrass vertices and divisor theory of graphs.
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Name(s): De Vas Gunasekara, Ajani Ruwandhika, Author
Brennan, Joseph, Committee Chair
Song, Zixia, Committee Member
Martin, Heath, Committee Member
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2018
Publisher: University of Central Florida
Language(s): English
Abstract/Description: 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.
Identifier: CFE0007397 (IID), ucf:52072 (fedora)
Note(s): 2018-05-01
M.S.
Sciences, Mathematics
Masters
This record was generated from author submitted information.
Subject(s): Divisor of a graph -- Dhar's burning algorithm -- gap sequence -- Weierstrass semigroup -- Weierstrass vertex -- Weierstrass weight
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0007397
Restrictions on Access: campus 2019-11-15
Host Institution: UCF

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