We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by F\"urer (Lin. Alg. Appl. 2010) based on the angles formed by the set of standard basis vectors and their projections onto eigenspaces of the adjacency matrix. We provide a purely combinatorial characterization of this hierarchy in terms of the walk counts. This allows us to give a complete answer to F\"urer's question about the strength of his invariants in distinguishing non-isomorphic graphs in comparison to the 2-dimensional Weisfeiler-Leman algorithm, extending the recent work of Rattan and Seppelt (SODA 2023). As another application of the characterization, we prove that almost all graphs are determined up to isomorphism in terms of the spectrum and the angles, which is of interest in view of the long-standing open problem whether almost all graphs are determined by their eigenvalues alone. Finally, we describe the exact relationship between the hierarchy and the Weisfeiler-Leman algorithms for small dimensions, as also some other important spectral characteristics of a graph such as the generalized and the main spectra.

翻译：我们考虑一类自然扩展了F\"urer（Lin. Alg. Appl. 2010）基于标准基向量集及其在邻接矩阵特征空间投影所形成角度定义的谱不变量的图不变量层级。本文给出了该层级关于游走计数的纯组合刻画，从而完整解答了F\"urer关于其不变量与二维Weisfeiler-Leman算法在区分非同构图方面强度比较的问题，延伸了Rattan与Seppelt（SODA 2023）的最新工作。作为该刻画的另一应用，我们证明了几乎所有图均可通过谱与角度在 isomorphism 意义下被唯一确定——这一结论对"几乎所有图是否仅由特征值唯一确定"这一长期未决问题具有重要启示。最后，我们给出了该层级与低维Weisfeiler-Leman算法之间，以及与其他重要谱特征（如广义谱与主谱）之间的精确关系。