On a Hierarchy of Spectral Isomorphism Invariants

We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by Fürer (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 charact...

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Bibliographic Details
Published in:Computational complexity Vol. 34; no. 2; p. 19
Main Authors: Arvind, V., Fuhlbrück, Frank, Köbler, Johannes, Verbitsky, Oleg
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.12.2025
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Combinatorial analysis
Computational Mathematics and Numerical Analysis
Computer Science
Eigenvalues
Graphs
Invariants
Isomorphism
combinatorial refinement
strongly regular graphs
68Q25
Graph isomorphism
68R10
spectra of graphs
ISSN:1016-3328, 1420-8954
Online Access:Get full text
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Summary:We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by Fürer (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 Fürer's question about the strength of his invariants in distinguishing non-isomorphic graphs in comparison with 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.
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ISSN:1016-3328
1420-8954
DOI:10.1007/s00037-025-00270-x