We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by F\"urer (Lin. Alg. Appl. 2010) based on angles between the projections of standard basis vectors onto an eigenspace of the adjacency matrix of a graph. 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 by their eigenvalues and angles, which is closely related to the long-standing open problem whether almost all graphs are determined by their spectrum. 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：我们考虑一种图不变量的层次结构，该结构自然地推广了Führer（《线性代数及其应用》2010年）基于图邻接矩阵特征空间上标准基向量投影之间的角度所定义的谱不变量。我们通过游走计数给出了该层次结构的纯组合刻画。这使得我们能够完全回答Führer关于其不变量与二维Weisfeiler-Leman算法在区分非同构图方面能力的问题，从而扩展了Rattan和Seppelt（SODA 2023）的最新工作。作为该刻画的另一个应用，我们证明了几乎所有图都可以通过其特征值和角度确定至同构，这与长期悬而未决的问题——即几乎所有图是否都可以通过其谱确定——密切相关。最后，我们描述了该层次结构与低维Weisfeiler-Leman算法之间的精确关系，以及图的其他重要谱特征（如广义谱和主谱）之间的关系。