Lando S. Polynomial graph invariants and the KP hierarchy (abstract)

We describe a large family of polynomial graph invariants whose average value is a $\displaystyle{ \tau }$-function for the Kadomtsev-Petviashvili hierarchy of partial differential equations. In particular, this is valid for Stanley's symmetrized chromatic polynomial, as well as for the Abel polynomial for graphs we introduce. The key point here is a Hopf algebra structure on the space spanned by graphs and the behavior of the invariants on its primitive space.