## Abstract

We prove that the generating function for the symmetric chromatic polynomial of all simple graphs is (after an appropriate scaling change of variables) a linear combination of one-part Schur polynomials. This statement immediately implies that it is also a τ-function of the Kadomtsev–Petviashvili integrable hierarchy of mathematical physics. Moreover, we describe a large family of polynomial graph invariants leading to the same τ-function. In particular, we introduce the Abel polynomial for graphs and show this for its generating function. 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.

Original language | English |
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Article number | 34 |

Journal | Selecta Mathematica, New Series |

Volume | 26 |

Issue number | 3 |

DOIs | |

Publication status | Published - 8 Jun 2020 |