Title: Applications of Linear Algebra and UPC Matroids
Abstract: Linear Algebra is a subject rich in applications. In this talk, we will discuss some of those applications and particularly how we can use linear algebra to help us solve an unsolved problem in graph theory. In 1973, Entringer posed the question: what simple graphs on n vertices have exactly one cycle of each length from 3 to n? These graphs are called uniquely pancyclic graphs or UPC graphs, and there are only seven known UPC graphs. To further study this idea, we will use matroids, which arise from the shared behaviors of vector spaces and graphs. In this way, we will determine properties that a matroid would need in order to be a UPC matroid and perhaps give rise to a UPC graph.