AI 中文总结
该研究探讨图稳定空间$\bar{\beta}_{g,G}$的删除-收缩性质，推导弦与伸缩方程，将$\boldsymbol{\beta}$类积分等用色多项式表示，还得到图的Crapo $\beta$不变量新公式并推广至高亏格。
AI 中文摘要
图稳定空间$\boldsymbol{\bar{\beta}}_{g,G}$参数化标记结点曲线，其允许的标记碰撞由图决定。我们研究$\bar{\beta}_{g,G}$上$\boldsymbol{\beta}$类的相交数，以及簇格罗滕迪克环中的类$[M_{g,G}]$。在这两种情形下，我们证明几何由底层图结构主导，表现为删除-收缩关系。作为推论，我们推导弦方程和伸缩方程，并将多族$\boldsymbol{\beta}$类积分用色多项式表示。我们还将任意域上$M_{0,G}$的格罗滕迪克类用色多项式表示，并将多种欧拉特征与组合量对应。过程中，我们得到图的Crapo $\beta$不变量的新公式，最后将这些关系推广到亏格1，并在色条件下推广到更高亏格。
英文摘要
Graphically stable spaces $\overline{\mathcal{M}}_{g,G}$ parametrize marked nodal curves whose permitted collisions of markings are determined by a graph. We study intersection numbers of $ψ$-classes on $\overline{\mathcal{M}}_{g,G}$, as well as the classes $[M_{g,G}]$ in the Grothendieck ring of varieties. In both settings, we show that the geometry is governed by an underlying graphical structure, expressed through deletion-contraction relations. As consequences, we derive string and dilaton equations and express several families of $ψ$-class integrals in terms of the chromatic polynomial. We also express the Grothendieck class of $M_{0,G}$ over an arbitrary field in terms of the chromatic polynomial and identify various Euler characteristics with combinatorial quantities. Along the way, we obtain a new formula for Crapo's $β$-invariant of graphs. Finally, we extend these relations to genus one and, under a chromatic condition, to higher genus.
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