Research & Projects
Research & Projects
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Domination of Two Families of Fractal Graphs
Domination of Two Families of Fractal Graphs
In an attempt to further the study of graph domination, Mr. Owen Zimmer proposed to study this on two families of graphs that arise when taking a limit of a sequence of graphs. Matt McClinton, Dr. Pamela E. Harris, and I provided mentoring and feedback.
Objectives
Objectives
Our primary objective included determining bounds on the domination number of each graph in the families of Sierpinski gasket graphs and the square fractal graphs. After this, we explore assisted domination on these graphs.
Results
Results
Reasonable (linear) upper and lower bounds were deduced for one family of graphs. For the other, a suspected optimal domination number was given and explored. When we approach these problems via assisted domination, exact assisted domination numbers are realized for several graphs.
Discrete Language Analysis (UWM, 2026)
Discrete Language Analysis (UWM, 2026)
Inspired by research performed by Pamela, Dr. Jen Elder (Missouri Western State University), and Anthony Simpson in 2024, I performed research alongside fellow graduate students Joakim Jakovleski and Matt McClinton during the spring semester and summer.
Objectives
Objectives
Our primary objective included the computation of discrete statistics, e.g., ascents, descents, ties, peaks, valleys, major index, excedances, mesas, canyons, and inversions, when applied to words as sequences of totally-ordered letters. We also wanted to produce this data in several formats, including b-spline interpolation plots, heatmaps, and histograms.
Results
Results
Matt, Joakim, and I wrote plenty of Python code utilizing the code that Anthony provided. This allowed us to integrate more language dictionaries and define more combinatorial statistics. Also provided is discussion of the similarities and differences between patterns observed in our figures. Finally, words of maximal statistic are included for novelty (did you know that in British English, the word with the greatest number of ties is, ironically, successlessness?) In June 2026, we wrote an abridged version of the paper, along with some surprising results we encountered. In August 2026, we published the full (241-page!) paper.
The Computation of Lean Numbers (UWM, 2025 - 2026)
The Computation of Lean Numbers (UWM, 2025 - 2026)
Continuing Dr. Ng's and my research on hypergraph coloring, Dr. Phil Chang, Dr. Pamela E. Harris, and I used Python and high-performance computation tools to calculate the lean numbers of thousands of hypergraphs.
Objectives
Objectives
Our primary objective included the computation of as many lean numbers as possible. Since these computations are at least as difficult as graph coloring, which is NP-Hard, we had a secondary aim to produce other necessary and sufficient conditions that would make such computations more efficient.
Results
Results
Dr. Chang and I laid out the groundwork for the code that helped us calculate many lean numbers. Dr. Harris and I proved several lemmas that made computations far more efficient. After calculating the lean numbers of several thousand hypergraphs, we also ran statistical analyses of the resulting data, which illuminated relationships between hypergraphs' lean numbers and their other features. You may find the arXiv preprint of the paper here.
Lean Numbers of Hypergraphs (UMN, 2018 - 2019)
Lean Numbers of Hypergraphs (UMN, 2018 - 2019)
A hypergraph is a generalization of a graph which allows each edge to contain any number of vertices. In order to explore lean colorings of mathematical knots, we may instead explore lean colorings of the much larger class of hypergraphs.
Objectives
Objectives
Dr. Peh Ng and I sought three things: 1. A general algorithm to determine the lean number of a given hypergraph, 2. Necessary and sufficient conditions for a hypergraph to have a particular lean number, and 3. Plenty of data about the lean numbers of many hypergraphs.
Results
Results
Through plenty of literature review, Dr. Ng and I determined a straightforward algorithm to determine the lean number of any hypergraph, and several conditions that considerably simplified calculations. I presented results of this investigation at Saint Benedict's and Saint John's University in Minnesota in May 2019 to other student researchers.
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