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Inverse eigenvalue problems for graphs

An online research community sponsored by the


American Institute of Mathematics, San Jose, California

organized by

Jane Breen, Mary Flagg, Jephian Lin, and Bryan Shader

This research community, sponsored by AIM and the NSF, begin in 2021 and and focuses on combinatorial and linear algebraic problems related to inverse eigenvalue problems related to graphs. The inverse eigenvalue problem for graphs (IEPG) is to determine if a given set of real numbers is the spectrum of a matrix with a given graph. This problem and related variants have been of interest for many years and recently developed tools have accelerated progress and opened up new lines of inquiry. Zero forcing (ZF) is a propagation process on graphs. ZF has become a fundamental concept because it represents a nice bridge between combinatorics and linear algebra. ZF has also arisen independently in quantum information processing, monitoring electrical power networks, and graph search algorithms.

The primary scientific goals of this AIM research community are to

  • use all the tools--recently developed matrix theoretic ones, combinatorial approaches developed for ZF and other tools from algebra and analysis--to make significant progress on the IEP-G; and
  • investigate new lines of zero forcing (ZF) and its variants.
This ARC is designed to bring a diverse group of researchers into the IEPG-ZF area and enhance collaboration and sense of community. In particular, the community will support graduate students, early career researchers, and faculty at primarily undergraduate institutes, by providing them a network of collaborators and opportunities to expand their research interests and establish successful research programs.

Collaborative research groups are used to attack problems related to the IEPG and ZF. This ARC operates on an approximately annual cycle of research group formation, beginning with moderated problem sessions to identify significant research problems. AIM-style voting is then used to form new research groups. The groups collaborate, usually for a year or more, write up their results, and submit them for publication. The time commitment associated with any one research group is limited.

If you would like to participate, please apply by filling out the on-line form. Applications are open to all, and we especially encourage women, underrepresented minorities, and researchers from primarily undergraduate institutions to apply.

Information about current and past activities, including upcoming events and papers published, can be found on the IEPG-ZF ARC website.

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