Weisfeiler--Leman and Graph Spectra

Two simple undirected graphs are cospectral if their respective adjacency matrices have the same multiset of eigenvalues. Cospectrality yields an equivalence relation on the family of graphs which is provably weaker than isomorphism. In this paper, we study cospectrality in relation to another well-studied relaxation of isomorphism, namely $k$-dimensional Weisfeiler-Leman ($k$-WL) indistinguishability. Cospectrality with respect to standard graph matrices such as the adjacency or the Laplacian matrix yields a strictly finer equivalence relation than $2$-WL indistinguishability. We show that individualising one vertex plus running $1$-WL already subsumes cospectrality with respect to all such graph matrices. Building on this result, we resolve an open problem of F\"urer (2010) about spectral invariants and strengthen a result of Godsil (1981) about commute distances. Looking beyond $2$-WL, we devise a hierarchy of graph matrices generalising the adjacency matrix such that $k$-WL indistinguishability after a fixed number of iterations can be captured as a spectral condition on these matrices. Precisely, we provide a spectral characterisation of $k$-WL indistinguishability after $d$ iterations, for $k,d \in \mathbb{N}$. Our results can be viewed as characterisations of homomorphism indistinguishability over certain graph classes in terms of matrix equations. The study of homomorphism indistinguishability is an emerging field, to which we contribute by extending the algebraic framework of Man\v{c}inska and Roberson (2020) and Grohe et al. (2022).