05C50 Online Seminar: Veronika Furst

The Inverse Eigenvalue Problem for Graphs (IEP-G) concerns determining all possible spectra of matrices in S(G), the set of real symmetric matrices described by a graph. Specific subproblems involve studying the maximum nullity or minimum rank, the ordered multiplicity list of distinct eigenvalues, or the minimum number of distinct eigenvalues of a graph. In this talk, we will share recent results on the last of these, known as the parameter q(G). In particular, we will investigate bounds on the number of edges of a graph through the “allows problem,” which asks what sparsity allows q(G) = 2, and the complementary “requires problem,” which asks what density requires q(G) = 2. We will answer the first question, present a characterization of certain regular graphs, and give some evidence in support of a conjectured answer to the second question.

Joint work with W. Barrett, E. Egolf, S. Fallat, F. Kenter, S. Nasserasr, B. Rooney, M. Tait, and H. van der Holst, and partially supported by NSF grant DMS-2331072.