Author ORCID Identifier
Date Available
7-24-2023
Year of Publication
2023
Degree Name
Doctor of Philosophy (PhD)
Document Type
Doctoral Dissertation
College
Arts and Sciences
Department/School/Program
Mathematics
First Advisor
Dr. Benjamin Braun
Abstract
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. First, we establish
formulas or bounds for the number of facets attained by families of sparse, connected graphs, and give conjectures concerning the maximum and minimum facet counts for more general families. We also consider the number of facets of SEPs arising from graphs generated by several random graph models and investigate a conjectured connection between facet counts for SEPs and clustering metrics on their underlying graphs.
Digital Object Identifier (DOI)
https://doi.org/10.13023/etd.2023.297
Recommended Citation
Bruegge, Kaitlin, "Geometric and Combinatorial Properties of Lattice Polytopes Defined from Graphs" (2023). Theses and Dissertations--Mathematics. 102.
https://uknowledge.uky.edu/math_etds/102
