The Ordered Multiplicity Inverse Eigenvalue Sequence Problem, Powers Of Graphs, And More!

Time: 9:30 -- 11: 00, October 01, 2025

Venue: Room 612, A6, Institute of Mathematics-VAST

Abstract: Given a matrix pattern, one may ask: "What sets of eigenvalues are possible over all such matrices?'' This problem is very hard! A mild relaxation of this question considers the multiplicity sequence instead of the exact eigenvalues themselves. For instance, ``Given an $n times n$ matrix pattern and an ordered partition $(m_1, ldots, m_q)$ of $n$, is there a matrix with that pattern 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 symmetric matrix patterns up to size $6 times 6$.

In this talk, we develop methods using combinatorial optimization on networks to approach this otherwise linear-algebraic problem. We apply many different ``zero forcing'' techniques to simultaneously bound on sums of various multiplicities. Not only can we verify the result above in a more straight-forward manner, but we apply our techniques to more domains including skew-symmetric matrices, nonnegative matrices, among others. This is joint work with Jephian C.-H. Lin (National Sun Yat-sen University, Taiwan).