  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.
