Weisfeiler-Leman, Graph Spectra, and Random Walks

Gaurav Rattan, Tim Frederik Seppelt
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
more » ... alised Laplacians that the edge colours produced by 2-WL determine commute distances.
doi:10.18154/rwth-2022-01183 fatcat:4aus5vlbn5eonp2ajenvg2df54