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On a Hierarchy of Spectral Invariants for Graphs

Authors V. Arvind , Frank Fuhlbrück , Johannes Köbler , Oleg Verbitsky

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Author Details

V. Arvind
  • The Institute of Mathematical Sciences (HBNI), Chennai, India
  • Chennai Mathematical Institute, India
Frank Fuhlbrück
  • Institut für Informatik, Humboldt-Universität zu Berlin, Germany
Johannes Köbler
  • Institut für Informatik, Humboldt-Universität zu Berlin, Germany
Oleg Verbitsky
  • Institut für Informatik, Humboldt-Universität zu Berlin, Germany

Funding

  • Verbitsky, Oleg: Supported by DFG grant KO 1053/8-2. On leave from the IAPMM, Lviv, Ukraine.

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V. Arvind, Frank Fuhlbrück, Johannes Köbler, and Oleg Verbitsky. On a Hierarchy of Spectral Invariants for Graphs. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.STACS.2024.6

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 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph algorithms
Keywords
  • Graph Isomorphism
  • spectra of graphs
  • combinatorial refinement
  • strongly regular graphs

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References

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