PRIMES: The Inverse Eigenvalue Problem for Graphs and Collaboration to Promote Inclusivity in Undergraduate Mathematics Education

Fort Lewis College and the American Institute of Mathematics will engage in a two-year partnership, aimed at furthering research in pure mathematics at Fort Lewis College in a way that increases both research output at a primarily undergraduate institution (PUI) and inclusivity among historically underrepresented (UR) students. This partnership will support the PRIMES goal of enabling, building, and growing collaboration between Fort Lewis College (FLC), a Minority Serving Institution, and the American Institute of Mathematics (AIM), a DMS-supported Mathematical Sciences Research Institute. By supporting research and outreach efforts at FLC, this project will not only advance research excellence at a PUI but also increase diversity and promote inclusiveness, given the highly diverse student body of FLC. FLC's historic mission is the education of American Indian and Alaska Native (AI/AN) student populations, and first-generation college students comprise nearly half of the student body. The partnership with AIM will focus on both research excellence and efforts that promote increased retention of first-year UR students, especially in the STEM disciplines, who may struggle both academically and with a sense of belonging in college. The mathematical focal area of the project is on subproblems of the broad Inverse Eigenvalue Problem for Graphs (IEPG), which asks to determine all possible spectra of matrices whose off-diagonal entries match the zero-pattern of the adjacency matrix of a graph. The IEPG is a difficult problem of interest to many graph theorists and combinatorial matrix theorists. The PI and her collaborators aim to add to the body of knowledge on the minimum number of distinct eigenvalues over symmetric matrices described by a graph, by expanding their previous characterization of regular graphs of degree at most four, possibly in several directions. In addition to considering eigenvalues, the PI and other collaborators investigate the sparsity of null vectors (and thereby eigenvectors) of matrices associated with a graph. The concept of the spark of a matrix (prevalent in the area of compressed sensing) is adapted to the spark of a graph. Through mentorship of undergraduate researchers, the connection between the minimum number of distinct eigenvalues of strongly regular graphs and finite frame theory is also explored. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.