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 \(\tau \)-function of the Kadomtsev–Petviashvili integrable hierarchy of mathematical physics. Moreover, we describe a large family of polynomial graph invariants leading to the same \(\tau \)-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.

中文翻译：

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多项式图不变式和KP层次

我们证明，所有简单图的对称色多项式的生成函数是（在变量进行适当的缩放更改之后）单部分Schur多项式的线性组合。该陈述立即暗示它也是Kadomtsev–Petviashvili数学物理的可积分层次的\（\ tau \） -函数。此外，我们描述了导致相同\（\ tau \） -函数的多项式多项式图不变量。特别是，我们为图形引入了Abel多项式，并对其生成函数进行了说明。这里的关键是图跨越的空间上的Hopf代数结构以及其原始空间上不变量的行为。