The Ordered Multiplicity Inverse Eigenvalue Sequence Problem, Powers Of Graphs, And More! 02/13/2023

Speaker: Franklin Kenter, USNA
Date and time: Monday, February 13, 4–5 pm
Place: Rome 206

Abstract: Given a graph G, one may ask: "What sets of eigenvalues are possible over all weighted adjacency matrices of G?'' (Here, negative and diagonal weights are allowed.) This is known as the Inverse Eigenvalue Problem for Graphs (IEPG) and is very hard! A mild relaxation of this question considers the multiplicity sequence instead of the exact eigenvalues themselves. For instance, given a graph G on n vertices and an ordered partition $\mathbf{m}= (m_1, \ldots, m_\ell)$ of n, is there a weighted adjacency matrix where the i-th distinct eigenvalue has multiplicity $m_i$ ? This is known as the ordered multiplicity inverse eigenvalue sequence problem. Recent work has solved this problem for all graphs on 6 vertices.

In this talk, we develop zero forcing methods for the ordered multiplicity IEPG in a multitude of different contexts. Namely, we apply a menagerie of zero forcing parameters on powers of graphs to achieve bounds on sums of various multiplicities. Not only can we verify the above result in a more straight-forward manner, but we apply our techniques to skew-symmetric matrices, nonnegative matrices, among others.

This is joint work with Jephian C.-H. Lin (National Sun Yat-sen University).