🤖 AI Summary
This work investigates the discriminative power of spectral angles—the angles between eigenvectors of the adjacency matrix and standard basis vectors—for graph isomorphism testing. Methodologically, it establishes a purely combinatorial characterization of spectral angles at the level of walk counts, enabling a precise comparison with the Weisfeiler–Leman (WL) hierarchy. The contributions are threefold: (i) it proves that spectral angles are strictly equivalent in expressive power to the 2-dimensional WL algorithm (2-WL), yet strictly weaker than 3-WL—thereby fully resolving an open problem posed by Fürer regarding this invariant; (ii) it uncovers intrinsic connections between spectral angles, generalized spectra, and principal spectra; and (iii) it demonstrates that “almost all graphs are uniquely determined by their spectrum together with spectral angles”, yielding significant progress toward the long-standing conjecture on spectral uniqueness of graphs.
📝 Abstract
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.
