Skip to main content
SpringerLink
Log in

On the Combinatorial Power of the Weisfeiler-Lehman Algorithm

  • Conference paper
  • First Online:
Algorithms and Complexity (CIAC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10236))

Included in the following conference series:

Abstract

The classical Weisfeiler-Lehman method WL[2] uses edge colors to produce a powerful graph invariant. It is at least as powerful in its ability to distinguish non-isomorphic graphs as the most prominent algebraic graph invariants. It determines not only the spectrum of a graph, and the angles between standard basis vectors and the eigenspaces, but even the angles between projections of standard basis vectors into the eigenspaces. Here, we investigate the combinatorial power of WL[2]. For sufficiently large k, WL[k] determines all combinatorial properties of a graph. Many traditionally used combinatorial invariants are determined by WL[k] for small k. We focus on two fundamental invariants, the number of cycles \(C_p\) of length p, and the number of cliques \(K_p\) of size p. We show that WL[2] determines the number of cycles of lengths up to 6, but not those of length 8. Also, WL[2] does not determine the number of 4-cliques.

M. Fürer—This work was partially supported by NSF Grant CCF-1320814. Part of this work has been done while visiting Theoretical Computer Science, ETH Zürich, Switzerland.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    A multiset differs from a set by assigning a positive integer multiplicity to each element.

References

  1. Alon, N., Yuster, R., Zwick, U.: Finding and counting given length cycles. Algorithmica 17(3), 209–223 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alzaga, A., Iglesias, R., Pignol, R.: Spectra of symmetric powers of graphs and the Weisfeiler-Lehman refinements. J. Comb. Theory Ser. B 100(6), 671–682 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babai, L.: Graph isomorphism in quasipolynomial time. CoRR, abs/1512.03547 (2015)

    Google Scholar 

  4. Babai, L.: Graph isomorphism in quasipolynomial time (extended abstract). In: Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pp. 684–697. ACM (2016)

    Google Scholar 

  5. Babai, L., Erdős, P., Selkow, S.M.: Random graph isomorphism. SIAM J. Comput. 9(3), 628–635 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  6. Babai, L., Kučera, L.: Graph canonization in linear average time. In: Proceedings of the 20th Annual Symposium on Foundations of Computer Science (FOCS), pp. 39–46. IEEE Computer Society Press (1979)

    Google Scholar 

  7. Babai, L., Mathon, R.: Talk at the South-East Conference on Combinatorics and Graph Theory (1980)

    Google Scholar 

  8. Cai, J.-Y., Fürer, M., Immerman, N.: An optimal lower bound on the number of variables for graph identification. Combinatorica 12(4), 389–410 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cameron, P.J.: 6-transitive graphs. J. Comb. Theory Ser. B 28(2), 168–179 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  10. Flum, J., Grohe, M.: The parameterized complexity of counting problems. SIAM J. Comput. 33(4), 892–922 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fürer, M.: Graph isomorphism testing without numerics for graphs of bounded eigenvalue multiplicity. In: Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 624–631 (1995)

    Google Scholar 

  12. Fürer, M.: On the power of combinatorial and spectral invariants. Linear Algebra Appl. 432(9), 2373–2380 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gol’fand, Y.Y., Klin, M.H.: On \(k\)-regular graphs. In: Algorithmic Research in Combinatorics, pp. 76–85. Nauka Publ., Moscow (1978)

    Google Scholar 

  14. Klin, M.C., Pöschel, R., Rosenbaum, K.: Angewandte Algebra. Vieweg & Sohn Publ., Braunschweig (1988)

    MATH  Google Scholar 

  15. Kučera, L.: Canonical labeling of regular graphs in linear average time. In: Proceedings of the 28th Annual Symposium on Foundations of Computer Science (FOCS), pp. 271–279. IEEE Computer Society Press (1987)

    Google Scholar 

  16. Luks, E.M.: Isomorphism of graphs of bounded valence can be tested in polynomial time. J. Comput. Syst. Sci. 25, 42–65 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  17. Read, R.C., Corneil, D.G.: The graph isomorphism disease. J. Graph Theory 1(4), 339–363 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  18. Weisfeiler, B. (ed.): On Construction and Identification of Graphs. Lecture Notes in Mathematics. Springer, Berlin (1976). With contributions by A. Lehman, G. M. Adelson-Velsky, V. Arlazarov, I. Faragev, A. Uskov, I. Zuev, M. Rosenfeld and B. Weisfeiler

    MATH  Google Scholar 

  19. Williams, V.V.: Multiplying matrices faster than Coppersmith-Winograd. In: Proceedings of the 44th Symposium on Theory of Computing Conference (STOC), pp. 887–898. ACM (2012)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, Pennsylvania State University, University Park, PA, USA

    Martin Fürer

Authors
  1. Martin Fürer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Fürer .

Editor information

Editors and Affiliations

  1. National Technical University of Athens, Zografou, Athens, Greece

    Dimitris Fotakis

  2. National Technical University of Athens, Zografou, Athens, Greece

    Aris Pagourtzis

  3. Université Paris-Dauphine, Paris, France

    Vangelis Th. Paschos

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Fürer, M. (2017). On the Combinatorial Power of the Weisfeiler-Lehman Algorithm. In: Fotakis, D., Pagourtzis, A., Paschos, V. (eds) Algorithms and Complexity. CIAC 2017. Lecture Notes in Computer Science(), vol 10236. Springer, Cham. https://doi.org/10.1007/978-3-319-57586-5_22

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-57586-5_22

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-57585-8

  • Online ISBN: 978-3-319-57586-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation