SIAM Review, Volume 62, Issue 3, Page 716-726, January 2020.
We present an elementary proof of a generalization of Kirchhoff's matrix tree theorem to directed, weighted graphs. The proof is based on a specific factorization of the Laplacian matrices associated to the graphs, which involves only the two incidence matrices that capture the graph's topology. We also point out how this result can be used to calculate principal eigenvectors of the Laplacian matrices.

中文翻译：

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有向图的矩阵树定理的初等证明

SIAM评论，第62卷，第3期，第716-726页，2020年1月。
我们提供了基尔霍夫矩阵树定理到有向加权图的推广的基本证明。该证明基于与图相关的拉普拉斯矩阵的特定因式分解，该拉普拉斯矩阵仅涉及捕获图拓扑的两个入射矩阵。我们还指出如何将此结果用于计算拉普拉斯矩阵的主要特征向量。