On a Hierarchy of Spectral Invariants for Graphs

TL;DR

This paper introduces a hierarchy of spectral invariants for graphs, providing combinatorial characterizations and comparing their distinguishing power to the Weisfeiler-Leman algorithm, with implications for graph isomorphism problems.

Contribution

It offers a combinatorial characterization of a spectral invariants hierarchy and analyzes its effectiveness in graph isomorphism testing compared to existing algorithms.

Findings

Almost all graphs are determined by spectrum and angles

Hierarchy extends F"urer's spectral invariants

Relationship established with Weisfeiler-Leman algorithms

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.