Weisfeiler--Leman and Graph Spectra

We devise a hierarchy of spectral graph invariants, generalising the adjacency spectra and Laplacian spectra, which are commensurate in power with the hierarchy of combinatorial graph invariants generated by the Weisfeiler--Leman (WL) algorithm. More precisely, we provide a spectral characterisation of $k$-WL indistinguishability after $d$ iterations, for $k,d \in \mathbb{N}$. Most of the well-known spectral graph invariants such as adjacency or Laplacian spectra lie in the regime between 1-WL and 2-WL. We show that individualising one vertex plus running 1-WL is already more powerful than all such spectral invariants in terms of their ability to distinguish non-isomorphic graphs. Building on this result, we resolve an open problem of F\"urer (2010) about spectral invariants and strengthen a result due to Godsil (1981) about commute distances.