We study the heat kernel expansion of the Laplacian on n-forms defined on a subgraph of a directed complete graph. We derive two expressions for the subgraph heat kernel on 0-forms and compute the coefficients of the expansion. We also obtain the subgraph heat kernel of the Laplacian on 1-forms.

中文翻译：

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在完整图的子图上定义的形式上的热核

我们研究了在有向完全图的子图上定义的n 型上拉普拉斯算子的热核扩展。我们推导出 0 型子图热核的两个表达式，并计算膨胀系数。我们还获得了 1 型上拉普拉斯算子的子图热核。