Date Available
4-25-2018
Year of Publication
2018
Degree Name
Doctor of Philosophy (PhD)
Document Type
Doctoral Dissertation
College
Arts and Sciences
Department/School/Program
Mathematics
First Advisor
Dr. Benjamin Braun
Abstract
Graphs provide interesting ways to generate families of lattice polytopes. In particular, one can use matrices encoding the information of a finite graph to define vertices of a polytope. This dissertation initiates the study of the Laplacian simplex, PG, obtained from a finite graph G by taking the convex hull of the columns of the Laplacian matrix for G. The Laplacian simplex is extended through the use of a parallel construction with a finite digraph D to obtain the Laplacian polytope, PD.
Basic properties of both families of simplices, PG and PD, are established using techniques from Ehrhart theory. Motivated by a well-known conjecture in the field, our investigation focuses on reflexivity, the integer decomposition property, and unimodality of Ehrhart h*-vectors of these polytopes. A systematic investigation of PG for trees, cycles, and complete graphs is provided, which is enhanced by an investigation of PD for cyclic digraphs. We form intriguing connections with other families of simplices and produce G and D such that the h*-vectors of PG and PD exhibit extremal behavior.
Digital Object Identifier (DOI)
https://doi.org/10.13023/ETD.2018.157
Recommended Citation
Meyer, Marie, "Polytopes Associated to Graph Laplacians" (2018). Theses and Dissertations--Mathematics. 54.
https://uknowledge.uky.edu/math_etds/54