Generalizing the $\star$‑graphs of Müller–Stach and Westrich, we describe a class of graphs whose associated graph hypersurface is equipped with a non-trivial torus action. For such graphs, we show that the Euler characteristic of the corresponding projective graph hypersurface complement is zero. In contrast, we also show that the Euler characteristic in question can take any integer value for a suitable graph. This disproves a conjecture of Aluffi in a strong sense.

中文翻译：

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绘制具有环面作用的超曲面和 Aluffi 猜想

概括 Müller–Stach 和 Westrich 的 $\star$-graphs，我们描述了一类图，其关联的图超曲面配备了非平凡的环面动作。对于这样的图，我们表明相应的投影图超曲面补集的欧拉特征为零。相比之下，我们还表明，所讨论的欧拉特征可以为合适的图形取任何整数值。这在很大程度上反驳了阿鲁菲的猜想。