Graph Polynomials and Invariants
American Institute of Mathematics, Pasadena, California
organizers:
Aihua Li and Sarita Nemani
This research community, sponsored by AIM and the NSF, will begin in 2026 and focus on the study of algebraic properties of graph invariants, especially on polynomials of certain graphs derived from scientific problems. Graph polynomials encode graph-theoretical information and useful graph parameters of the underlying graph in various ways. The proposed research will integrate advanced techniques from combinatorics and algebra to develop graphical properties of the selected graphs. In particular, when the graphs represent certain networks, the study has the potential in helping the understanding of the stability of the networks and in the development of the mathematical models.
The primary scientific goals of this AIM research community are to investigate the algebraic properties of several types of graph polynomials, such as interlace polynomials and Hosoda polynomials of selected graphs, and identify graph theory properties derived from these polynomials. Effort will be made to investigate the less-touched 2-variable generalized interlace polynomials. Algebraic and combinatorial techniques such as operations on adjacency matrices and searching zeros of a polynomial, and the matroid method will be applied in the study.
This ARC is designed to bring a diverse group of researchers into the area of graph polynomials and invariants 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 such as
- How the graph polynomials in the study can tell important properties of the underlying graphs?
- Is there any connection in between the polynomial values or coefficients and the connectivities, clustering situation, closeness of the graph, and other related graph theory properties?
This ARC will begin in February 2026 with two introductory lectures by experts in the field, followed by moderated problem sessions to identify significant research problems. Research groups will be formed and will 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 by February 10. Applications are open to all, and we especially encourage researchers from primarily undergraduate institutions to apply. Information about current activities can be found on community website.
If you have questions about the program, please contact us at graphpolynomials@gmail.com .
