Weisfeiler--Leman, Graph Spectra, and Random Walks

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.