Speaker
Description
An inverse eigenvalue problem for a graph (IEP-G) asks a fundamental question: What are the possible spectra for (symmetric) real matrices fitting a given graph? Many have worked on several aspects of the IEP-G with exciting advances and variations appearing over the past forty years. Here, the focus will be on weighted Laplacian matrices associated with a graph. Such matrices are permanently intertwined in combinatorial matrix theory and numerical analysis and form a natural matrix class to study for an inverse eigenvalue problem.
For a given graph G, we aim to determine the possible realizable spectra for a generalized (or weighted) Laplacian matrix associated with G. We present such results for certain families of graphs and graphs on a small number of vertices, including possible ordered multiplicity lists.
Strong matrix properties have been associated with several adaptations of inverse eigenvalue problems and sign pattern eigenvalue problems. We introduce a new strong property, the strong spectral property for weighted Laplacian matrices (SSPWL), and establish its Supergraph and Bifurcation lemmas. We develop a Jacobian method which can be used to verify such a strong property. These tools are then applied to derive potential spectral regions of weighted Laplacian matrices of a graph on four vertices.