Atlantic Graph Theory Seminar

The Martin Invariant and Other Results on the Interlace Polynomials

Josephine Reynes, University of Waterloo

There are many well studied graph polynomials, but this talk will focus on the Martin polynomial and the interlace polynomial. Specifically, this talk will look at how these two polynomials are related and how results on the Martin polynomial can be extended to the interlace polynomial. The Martin invariant, a specific evaluation of the Martin polynomial, obeys the symmetries of the Feynman period. The Feynman period of a graph is useful in quantum field theory, but difficult to compute and thus there is interest in finding graph invariants that have the same symmetries. It was quickly established that the interlace polynomial on interlace graphs was equal to the Martin polynomial on the associated 4-regular graph. While only graphs not containing a set of forbidden vertex minors are interlace graphs, the interlace polynomial is defined over all graphs. We discuss how this provides a way to try and extend the notion of Feynman symmetries via the interlace polynomial and some specific classes of graphs with formulas. Additionally, the interlace polynomial is only equal to the Martin polynomial for interlace graphs of 4-regular graphs, but the Martin polynomial is defined for 2k-regular graphs. Thus, we work toward creating an interlace-like polynomial for graphs derived from 2k-regular cases of the Martin polynomial.

Zoom link:
https://us02web.zoom.us/j/86861499971?pwd=rTDAaju0TCu24asnaBGvkuNlT11KZ1.1