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May 17, 2017

Flow and Yamada Polynomials of Cubic Graphs

We?ll discuss Tutte?s golden identity for flow polynomials of planar cubic graphs, and an extension to Yamada polynomials of spatial cubic graphs. This explains a curious property of flow polynomials of cubic graphs (mod 5). We conjecture that the golden identity becomes an inequality for non-zero flow polynomials of general cubic graphs which characterizes planarity, and have proved this for certain infinite classes of cubic (non-planar) graphs. We?ll introduce the chromatic algebra to help explain these phenomena, and use it to show that the number of flow polynomials of planar cubic graphs grows exponentially with the degree, answering a question of Treumann-Zaslow. This is joint work with Slava Krushkal.