Speaker
Description
The Inverse Eigenvalue Problem for a Graph (IEP-G) asks for the possible spectra of a real symmetric matrix knowing only which off-diagonal entries are non-zero, as described by a graph $G$. Three matrix properties, collectively called the “strong properties”, have become prominent in the study of this problem, due in part to their good behavior with respect to edge deletion and contraction within $G$.
The three properties differ in what eigenvalue data is preserved:
- For the Strong Arnold Property (SAP), a single multiplicity.
- For the Strong Multiplicity Property (SMP), all multiplicities.
- For the Strong Spectral Property (SSP), all multiplicities and locations.
In most cases, a question that calls for a strong property will be served well by one of these three perspectives. There does, however, exist a more general perspective that contains each of these as a special case, here introduced as the General Strong Property (GSP). For the GSP, any chosen subset of eigenvalue multiplicities can be preserved, and eigenvalue locations can be allowed to vary locally subject to any set of linear constraints. The usual strong property consequences follow, which include: an algebraic definition, a verification matrix, superpatterns, edge decontraction, matrix liberation, and bifurcation.
In addition, the null space characterization of SAP given by H. van der Holst is extended to an eigenspace characterization of GSP, including the special cases of SMP and SSP, whose eigenspace characterization has not appeared in the literature.