The Weisfeiler--Leman algorithm is a ubiquitous tool for the Graph Isomorphism Problem with various characterisations in e.g. descriptive complexity and convex optimisation. It is known that graphs that are not distinguished by the two-dimensional variant have cospectral adjacency matrices. We tackle a converse problem by proposing a set of matrices called Generalised Laplacians that characterises the expressiveness of WL in terms of spectra. As an application to random walks, we show using Generalised Laplacians that the edge colours produced by 2-WL determine commute distances.

中文翻译：

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Weisfeiler-Leman，图谱和随机游走

Weisfeiler-Leman算法是一种用于图同构问题的无处不在的工具，它具有各种特征，例如描述复杂性和凸优化。众所周知，未被二维变量区分的图具有共谱邻接矩阵。我们通过提出一组称为广义Laplacians的矩阵来解决一个相反的问题，该矩阵以光谱的形式表征WL的表达性。作为随机游走的一种应用，我们使用广义拉普拉斯算子证明了2-WL产生的边缘颜色决定了通勤距离。