Author ORCID Identifier

https://orcid.org/0009-0007-3425-3216

Date Available

5-9-2025

Year of Publication

2025

Document Type

Doctoral Dissertation

Degree Name

Doctor of Philosophy (PhD)

College

Arts and Sciences

Department/School/Program

Mathematics

Faculty

Dr. Benjamin Braun

Abstract

This dissertation addresses two distinct problems related by their foundation in network flows. The first problem concerns volumes of flow polytopes of directed acyclic graphs with out-degree sequence (3,2,...,2,0). It is proved that there is an interchange operation on the edge set of these graphs that induces a partial order on the graphs isomorphic to a Boolean algebra, and that moving up through this partial order decreases (weakly) the volumes of the corresponding flow polytopes. This result is reinterpreted in the context of linear extensions for posets that are bipartite non-crossing trees.

The second problem develops a discrete optimization model for Lexington, Kentucky's bicycle network. Riding bikes is a crucial piece of a larger transportation sustainability puzzle, but many car-centric cities have incomplete bike networks. How can cities identify locations for new bicycle facilities that will achieve the lowest average travel distance for the population? Cyclist experience of safety and comfort has been quantified as ``stress'' based on road characteristics. To identify impactful additions to low-stress networks, a network flow optimization model is introduced that minimizes average trip distance within a city. Using solver software and Lexington, Kentucky, USA as a case study, near-optimal solutions to the were found for different input parameters. These results can inform planning decisions in Lexington, and the method can be carried out for any other city for which data is available.

Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2025.61

Funding Information

This work was partially supported by National Science Foundation award DMS-1953785 (2023-2024).

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