05C50 Online Seminar: Helena Šmigoc

The Inverse Eigenvalue Problem for a Graph (IEPG) asks for all possible spectra of real symmetric matrices whose pattern of off-diagonal nonzero entries is specified by a given graph. A subproblem to the IEPG is to determine all possible ordered multiplicity lists of eigenvalues. Once an ordered multiplicity list m is known to be realisable for a graph G, one needs to determine the set of all possible assignments of eigenvalues for m. If every set of distinct eigenvalues can be assigned to m, then we say that m is spectrally arbitrary for G. We present an infinite family of trees and ordered multiplicity lists whose sets of realising eigenvalues are highly constrained. Furthermore, we exhibit examples where multiplicity lists can only be achieved by a unique (up to shifting and scaling) set of eigenvalues.

This talk is based on joint work with Shaun M. Fallat, H. Tracy Hall, Rupert H. Levene, Seth A. Meyer, Shahla Nasserasr and Polona Oblak.