Polynomial graph invariants and the KP hierarchy

TL;DR

This paper demonstrates that the generating function for symmetric chromatic polynomials of connected graphs satisfies the KP hierarchy, revealing deep connections between graph invariants, integrable systems, and Hopf algebra structures.

Contribution

It establishes that certain graph invariants' generating functions solve the KP hierarchy and introduces the Abel polynomial for graphs within this framework.

Findings

Generating function for symmetric chromatic polynomial satisfies KP hierarchy

Introduction of Abel polynomial for graphs and its generating function

Hopf algebra structure underpins the invariants' behavior

Abstract

We prove that the generating function for the symmetric chromatic polynomial of all connected graphs satisfies (after appropriate scaling change of variables) the Kadomtsev--Petviashvili integrable hierarchy of mathematical physics. Moreover, we describe a large family of polynomial graph invariants giving the same solution of the KP. 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.