**Research with Students**

what I have done and what I am interested in

I love researching with students, and this page is dedicated to research I have done with students in the past, research I am currently doing with students, and problems I am interested in. I especially love research problems in the areas of algebra, combinatorics, and graph theory. If you have a problem you are interested in, please come tell me about it!

**Rainbow labelings**** with Dr. Kristen Barnard**** and J****akob Bister**

Jakob developed the concept of a rainbow labeling of the sequence of integers 0, 1, 2, . . . , n. Furthermore, he showed that every such rainbow labeling corresponds to a graceful labeling of a caterpillar graph. Conversely, it can be shown that every caterpillar graph has a graceful labeling stemming from a rainbow labeling.

**Graceful cacti**** with Dr. Kristen Barnard**** and**** Kaimera Greear**

Kaimera considered a class of graphs known as cactus graphs. In this project, they were able to show that several classes of cactus graphs are graceful. It should be noted that cactus graphs are not trees, so this doesn't fall under the graceful tree conjecture.

**Properties of matrix multiplication with graphs**** with Dr. Kristen Barnard and Nelson Xunic Cua**

Nelson explored two types of matrices corresponding to a graph, the adjacency matrix *A *and the incidence matrix *E*. He then considered the product (*A^n)E* and is able to explain the significance of this product relative to the graph.

**Maintaining a graceful labeling of a graph with swapping labelings**** with Dr. Kristen Barnard**** and ****Orion Musselman**

Orion looked at when swapping two vertex labelings on a gracefully labeled graph yields another graceful labeling of the graph. In particular, he developed some criterion for when this is possible for particular classes of graphs.

**Collatz conjecture with Libby Lees**

Libby and I are looking at the famous Collatz Conjecture. It deals with a special piecewise function that takes a natural number and outputs half of that number, if the original number was even, or three times that number plus one, if the original number was odd. If you repeatedly perform this function, the conjecture is that as long as you start with a natural number, you will always get an output of one. For more information on the Collatz Conjecture go here .

**Domino tilings with Williem Rizer**

Williem and I looked at studying ways to count domino tilings in an initiative to make a more intuitive formula to count the number of domino tilings of a rectangular grid (replacing Kastelyn's formula, which involves transcendental functions). In this thesis, Williem was able to show each tiling of a rectangular board is uniquely determined by the arrangement of the dominoes on the doubly-even positions. Williem's master's thesis is viewable here .

**Graceful tree conjecture with Patrick Cone**

Patrick and I worked on the graceful tree conjecutre which was originally posed in 1963. The conjecture states that you can label the vertices of a tree with the integers 0 up to m in such a way that the labeling of the edges via assignment of the absolute value of the difference of the labels of the vertices causes the edges to be labeled by the integers 1 up to m. Patrick and I were able to develop the idea of an *adjacency diagram* to study whether a tree is graceful, and we were able to use these diagrams to prove that several classes of trees are graceful. The paper is published in the Electronic Proceedings of Undergraduate Math Day at the University of Dayton, check it out here .

**Magic polygons with Victoria Jakicic**

Victoria and I were to generalize the idea of a magic square to that of a magic polygon. Moreover, we were able to classify the set of magic polygons, as well as to determine several properties of these magic polygons (namely the magic sum as well as the center number). The paper is published in the Electronic Proceedings of Undergraduate Math Day at the University of Dayton, check it out here .

**Representations of real numbers as finite sums with Lewis Dominguez**

Lewis and I looked at a problem that was posed in Mathematics Magazine. The problem asked whether it was possible to represent a real number by a finite sum of elements in an open subset of the real numbers that contained one positive and one negative number. We were able to answer the question in the affirmative, as well as expand the problem to a statement for elements of the real plane. The paper is published in the Electronic Proceedings of Undergraduate Math Day at the University of Dayton, check it out here .

**The double-critical conjecture with Derek Hanely**

Derek and I looked at the double-critical conjecture, which was first posed in 1966. The conjecture states that the only simple, connected graphs that are double-critical are the complete graphs. In this thesis, Derek developed some algorithms in SageMath/CoCalc to study the double-critical conjecture, as well as determined that any graph with a path graph on four vertices as an induced subgraph is not double-critical. Derek's master's thesis is viewable here .