**Undergraduate #93**

**Discipline:**Mathematics and Statistics

**Subcategory:**Mathematics and Statistics

**Session:**1

**Room:**Tyler

**Roman Vasquez**

**- University of Central Florida**

**Co-Author(s):**Rachel Wofford, Whitworth University, Spokane, Washington; Dr. Steven Schlicker, Grand Valley State University, Allendale, Michigan

The Hausdorff metric provides a way to measure the distance between sets.; we use this metric to define a geometry. In this geometry, we say that a finite configuration [A,B] is a pair of disjoint finite sets A and B that satisfy certain distance conditions. The sets A and B form the endpoints of a line segment in this geometry. These line segments have many interesting properties worth studying. One specific property is that for a given a positive integer n, there is usually a configuration [A,B] for which there are n different sets on the line segment defined by [A,B] at every distance from A. We denote this number of sets by #([A,B]). We will explain how we can study these configurations using techniques from graph theory. Each configuration [A,B] can be identified with a bipartite graph G, and #([A,B]) is equal to the number of edge coverings of G. In this way, each bipartite graph G defines a line segment in the space of sets. We will share results about #(G) for different types of bipartite graphs, which then determines the number of points at each distance on the line segment corresponding to the graph G. This research was conducted as part of the 2019 REU program at Grand Valley State University.

**Funder Acknowledgement(s):** This work was partially supported by National Science Foundation grant DMS-1659113, which funds a Research Experiences for Undergraduates program at Grand Valley State University.

**Faculty Advisor: **Dr. Steven Schlicker,
schlicks@gvsu.edu

**Role:** I worked together with Rachel Wofford to find formulas to describe the 'num's of different kinds of finite configurations, which we were then able to prove. In doing so, we also discovered several never before seen integer sequences.