Inverse eigenvalue problems appear in various contexts throughout mathematics and engineering, and refer to determining all the lists of eigenvalues (spectrum) that are possible for a matrix fitting some description. The Inverse Eigenvalue Problem of a Graph (IEPG) refers to determining the possible spectra of real symmetric matrices whose pattern of nonzero off-diagonal entries is described by the edges of a given graph. This problem and related variants have been of interest for many years and were originally approached through the study of ordered multiplicity lists. It was thought by many researchers in the field that at least for a tree $T$, determining the ordered multiplicity lists of $T$ would suffice to determine the spectra of matrices described by $T$. When it was shown in 2004 that this was not the case, the focus of much of the research in the area shifted to the narrower question of maximum multiplicity, or equivalently maximum nullity or minimum rank of matrices described by the graph. One of the tools used for this work is the Strong Arnold Property, which characterizes having the constant rank manifold and constant pattern manifold intersect transversally at a matrix $A$. This mini-symposium will present recent developments that offer hope for progress on the Inverse Eigenvalue Problem of a Graph, including generalizations of the Strong Arnold Property, and applications of these ideas to the IEPG.
Monday 11/07, 16:00–18:00
- Mon 16:00–16:30, Room AV 04.17Generalizing the Strong Arnold Property from nullity to spectra: the Strong Spectral Property and Strong Multiplicity Property, H. Tracy Hall (Brigham Young University).
- Mon 16:30–17:00, Room AV 04.17On the minimum number of distinct eigenvalues of a graph, Shaun Fallat (University of Regina).
- Mon 17:00–17:30, Room AV 04.17Using a new zero forcing process to guarantee the Strong Arnold Property, Jephian C.-H. Lin (Iowa State University).
- Mon 17:30–18:00, Room AV 04.17Using the Jacobian method to solve several inverse eigenvalue problems for graphs, Keivan Hassani Monfared (University of Calgary).
Tuesday 12/07, 10:30–12:30
- Tue 10:30–11:00, Room AV 04.17Positive Semidefinite Matrix Completion, Universal Rigidity and the Strong Arnold Property, Antonios Varvitsiotis (Nanyang Technological University, Singapore).
- Tue 11:00–11:30, Room AV 04.17On the Northeast Property of Signed Graphs with Loops, Hein van der Holst (Georgia State University).
- Tue 11:30–12:00, Room AV 04.17Two-connected signed graphs with maximum nullity at most two, Marina Arav (Georgia State University).
- Tue 12:00–12:30, Room AV 04.17Results in $k$-forcing., Michael Young (Iowa State University).