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. 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).

翻译：两个简单无向图称为同谱图，如果它们的邻接矩阵具有相同的特征值多重集。同谱性在图的族上定义了一个等价关系，该关系被证明严格弱于同构。本文研究同谱性与另一种被广泛研究的同构松弛——即k维Weisfeiler-Leman（k-WL）不可区分性——之间的关系。相对于标准图矩阵（如邻接矩阵或拉普拉斯矩阵）的同谱性，产生了一个比2-WL不可区分性严格更细的等价关系。我们证明，对单个顶点进行个性化处理并执行1-WL已经足以涵盖相对于所有这些图矩阵的同谱性。基于这一结果，我们解决了Fürer（2010）关于谱不变量的一个开放问题。超越2-WL，我们设计了一个推广邻接矩阵的图矩阵层次结构，使得固定迭代次数后的k-WL不可区分性可以表示为这些矩阵上的谱条件。具体而言，我们给出了k-WL不可区分性在d次迭代后的谱刻画，其中k,d ∈ ℕ。我们的结果可以被视为通过矩阵方程对某些图类上的同态不可区分性的刻画。同态不可区分性的研究是一个新兴领域，我们通过扩展Mančinska和Roberson（2020）以及Grohe等人（2022）的代数框架为此领域做出贡献。