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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Veröffentlicht in:	Computational complexity Jg. 34; H. 2; S. 19
Hauptverfasser:	Arvind, V., Fuhlbrück, Frank, Köbler, Johannes, Verbitsky, Oleg
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Cham Springer International Publishing 01.12.2025 Springer Nature B.V
Schlagworte:	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
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Abstract	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.
AbstractList	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.
ArticleNumber	19
Author	Köbler, Johannes Verbitsky, Oleg Arvind, V. Fuhlbrück, Frank
Author_xml	– sequence: 1 givenname: V. surname: Arvind fullname: Arvind, V. organization: The Institute of Mathematical Sciences (HBNI) and Chennai, Mathematical Institute – sequence: 2 givenname: Frank surname: Fuhlbrück fullname: Fuhlbrück, Frank organization: Institut für Informatik, Humboldt Universität zu Berlin – sequence: 3 givenname: Johannes surname: Köbler fullname: Köbler, Johannes organization: Institut für Informatik, Humboldt Universität zu Berlin – sequence: 4 givenname: Oleg surname: Verbitsky fullname: Verbitsky, Oleg email: oleg.verbitsky@hu-berlin.de organization: Institut für Informatik, Humboldt Universität zu Berlin
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Cites_doi	10.1007/s10801-021-01065-3 10.37236/209 10.1016/0024-3795(93)90491-6 10.1016/j.ejc.2011.03.012 10.1137/0209047 10.1017/CBO9780511801518 10.1007/978-1-4613-8354-3_5 10.1016/S0195-6698(03)00100-8 10.1007/s00224-020-09990-9 10.1016/j.laa.2009.07.019 10.1017/CBO9781139086547 10.1145/2897518.2897542 10.1016/j.laa.2006.07.017 10.1002/jgt.20461 10.1016/0095-8956(80)90058-1 10.1016/S0024-3795(03)00483-X 10.1137/15M1049622 10.1090/S0273-0979-06-01126-8 10.1137/1.9781611977554.ch87 10.1007/BF01305232 10.1016/0024-3795(82)90104-5
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Combinatorial analysis Computational Mathematics and Numerical Analysis Computer Science Eigenvalues Graphs Invariants Isomorphism
Title	On a Hierarchy of Spectral Isomorphism Invariants
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