Polynomial graph invariants and the KP hierarchy

Sergei Chmutov ^[1] ; Maxim Kazarian ^[2] ; Sergei Lando ^[2]
1. [1] Ohio State University
  
  Ohio State University
  
  City of Columbus, Estados Unidos
2. [2] Higher School of Economics, National Research University
  
  Higher School of Economics, National Research University
  
  Rusia
Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 26, Nº. 3, 2020
Idioma: inglés
DOI: 10.1007/s00029-020-00562-w
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Resumen
- 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.