ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

## 20th ILAS Conference

In minisymposium: Generalizations of the Strong Arnold Property and the Inverse Eigenvalue Problem of a Graph

Mon 16:00–16:30, Room AV 04.17
Generalizing the Strong Arnold Property from nullity to spectra: the Strong Spectral Property and Strong Multiplicity Property
H. Tracy Hall (Brigham Young University)
Joint work with Wayne Barrett (Brigham Young University); Shaun Fallat (University of Regina); Leslie Hogben (Iowa State University); Jephian C.-H. Lin (Iowa State University); Bryan L. Shader (University of Wyoming)

The Strong Arnold Property has been used with great success in developing linear-algebraic graph parameters that count maximum nullity in a well-behaved way, starting with the parameter $\mu(G)$ introduced by Y. Colin de Verdière. In particular, the parameters so constructed are minor-monotone: They do not increase upon contraction of a subgraph. We generalize the Strong Arnold Property in such a way as to facilitate its use in graph parameters that depend on more than just the multiplicity of zero as an eigenvalue: The Strong Spectral Property provides for a matrix whose entire spectrum is robust with respect to small perturbations in the non-edges, and the Strong Multiplicity Property does the same for the ordered list of eigenvalue multiplicities.