Polynomial graph invariants and the KP hierarchy

Sergei Chmutov, Maxim Kazarian, Sergei Lando

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

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 languageEnglish
Article number34
JournalSelecta Mathematica, New Series
Volume26
Issue number3
DOIs
Publication statusPublished - 8 Jun 2020

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