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：我们研究一类图不变量层次结构，它自然地扩展了F\"urer (Lin. Alg. Appl. 2010) 基于标准基向量集与它们在邻接矩阵特征空间上投影所成夹角定义的谱不变量。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F\"urer关于其不变量在区分非同构图方面相对于二维Weisfeiler-Leman算法强度的问题，从而扩展了Rattan和Seppelt (SODA 2023) 的最新工作。作为该刻画的另一应用，我们证明了几乎所有图都可以通过谱和夹角信息在同构意义下被唯一确定，这对于理解"是否几乎所有图都能仅由其特征值确定"这一长期悬而未决的问题具有重要意义。最后，我们精确描述了该层次结构与低维Weisfeiler-Leman算法的关系，以及其与图的广义谱、主谱等其他重要谱特征的联系。