S. V. Chmutov, M. E. Kazarian, and S. K. Lando have recently introduced a class of graph invariants, which they called shadow invariants (these invariants are graded homomorphisms from the Hopf algebra of graphs to the Hopf algebra of polynomials in infinitely many variables). They proved that, after an appropriate rescaling of the variables, the result of the averaging of almost every such invariant over all graphs turns into a linear combination of single-part Schur functions and, thereby, becomes a τ-function of an integrable Kadomtsev-Petviashvili hierarchy. We prove a similar assertion for the Hopf algebra of framed graphs. At the same time, we show that there is no such an analogue for a number of other Hopf algebras of a similar nature, in particular, for the Hopf algebras of weighted graphs, chord diagrams, and binary delta-matroids. Thus, it turns out that the Hopf algebras of graphs and framed graphs are distinguished among the graded Hopf algebras of combinatorial nature.

中文翻译：

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框架图的不变性和Kadomtsev-Petviashvili层次结构

SV Chmutov，ME Kazarian和SK Lando最近引入了一类图不变式，它们称为阴影不变式（这些不变式是从图的Hopf代数到多项式的Hopf代数的无级变分的同态）。他们证明，在对变量进行适当的重新缩放后，所有图上几乎所有此类不变量的平均结果变成了单部分Schur函数的线性组合，从而变成了τKadomtsev-Petviashvili层次结构的功能。对于框架图的Hopf代数，我们证明了类似的断言。同时，我们表明，对于许多其他性质相似的霍普夫代数，尤其是加权图，弦图和二元δ-拟阵的霍普夫代数，没有类似的东西。因此，事实证明，图和框架图的霍普夫代数在组合性质的渐变霍普夫代数中是有区别的。