“Arithmetical structures on bidents,” a paper co-authored by Dr. Joel Louwsma, assistant professor of mathematics at Niagara University, was recently published in Discrete Mathematics, a journal that provides a common forum for significant research in many areas of discrete mathematics and combinatorics.
In the paper, Louwsma and his co-authors, Kassie Archer, Abigail C. Bishop, Alexander Diaz-Lopez, Luis D. García Puente, and Darren Glass, study arithmetical structures and their critical groups on bidents, which are graphs consisting of a path with two “prongs” at one end. They give a procedure for determining the number of arithmetical structures on the bident with n vertices and show that this number grows at the same rate as the Catalan numbers as n increases. They also completely characterize the groups that occur as critical groups of arithmetical structures on bidents.
“Our paper is one of the first publications about arithmetical structures, and I am eager to see how this area develops in the future,” Louwsma said. “Some of our techniques might apply more generally to graphs with more prongs or longer prongs, and we are exploring this now. Several other researchers have recently begun studying arithmetical structures, and there is potential for rapid development in the coming years. It is an exciting time to be working in this area.”
Louwsma began his work on the topic in June 2017, as a participant in the Research Experiences for Undergraduate Faculty program at the Institute for Computational and Experimental Research in Mathematics at Brown University. He has since published a paper with an undergraduate student on a related topic.