Skip to main content
SpringerLink
Log in

On the Eigenvalues of Grassmann Graphs, Bilinear Forms Graphs and Hermitian Forms Graphs

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

Motivated by conjectures of Karloff, and Van Dam and Sotirov on the smallest eigenvalues of Johnson and Hamming graphs, Brouwer, Cioabă, Ihringer and McGinnis obtained some new results involving the eigenvalues of various graphs coming from association schemes and posed some conjectures related to the eigenvalues of Grassmann graphs, Bilinear forms graphs and Hermitian forms graphs. In this paper, we prove some of their conjectures.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The details of these calculations can be found on the webpage https://github.com/Himanshugupta23/Hermitian-Graph-Eigenmatrix.

References

  1. Alon, N., Sudakov, B.: Bipartite subgraphs and the smallest eigenvalue. Comb. Probab. Comput. 9(1), 1–12 (2000)

    Article  MathSciNet  Google Scholar 

  2. Alon, N., Sudakov, B., Zwick, U.: Constructing worst case instances for semidefinite programming based approximation algorithms. SIAM J. Discret. Math. 15(1), 58–72 (2001/2002)

  3. Brouwer, A.E., Cioabă, S.M., Ihringer, F., McGinnis, M.: The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance regular-graphs with classical parameters. J. Comb. Theory Ser. B 133, 88–121 (2018)

    Article  MathSciNet  Google Scholar 

  4. Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, Heidelberg (1989)

    Book  Google Scholar 

  5. Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Springer, Berlin (2012)

    Book  Google Scholar 

  6. van Dam, E.R., Sotirov, R.: New bounds for the max-\(k\)-cut and chromatic number of a graph. Linear Algebra Appl. 488, 216–234 (2016)

    Article  MathSciNet  Google Scholar 

  7. Delsarte, P.: Association schemes and t-designs in regular semilattices. J. Comb. Theory Ser. A 20(2), 230–243 (1976)

    Article  MathSciNet  Google Scholar 

  8. Delsarte, P.: Bilinear forms over a finite field, with applications to coding theory. J. Comb. Theory Ser. A 25(3), 226–241 (1978)

    Article  MathSciNet  Google Scholar 

  9. Dumer, I., Kapralova, O.: Spherically punctured biorthogonal codes. IEEE Trans. Inf. Theory 59(9), 6010–6017 (2013)

    Article  MathSciNet  Google Scholar 

  10. Eisfeld, J.: The eigenspaces of the Bose–Mesner algebras of the association schemes corresponding to projective spaces and polar spaces. Des. Codes Cryptogr. 17(1–3), 129–150 (1999)

    Article  MathSciNet  Google Scholar 

  11. Godsil, C., Meagher, K.: Erdős–Ko–Rado Theorems: Algebraic Approaches. Cambridge University Press, Cambridge (2018)

    MATH  Google Scholar 

  12. Godsil, C., Royle, G.: Algebraic Graph Theory. Graduate Texts in Mathematics, vol. 207. Springer, New York (2001)

    Book  Google Scholar 

  13. Goemans, M., Williamson, D.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. Assoc. Comput. Mach. 42(6), 1115–1145 (1995)

    Article  MathSciNet  Google Scholar 

  14. Ihringer, F., Metsch, K.: Large \(\{0,1,\ldots , t\}\)-cliques in dual polar graphs. J. Comb. Theory Ser. A 154, 285–322 (2018)

    Article  MathSciNet  Google Scholar 

  15. Karloff, H.: How good is the Goemans–Williamson MAX CUT algorithm? SIAM J. Comput. 29(1), 336–350 (1999)

    Article  MathSciNet  Google Scholar 

  16. Schmidt, K.-U.: Hermitian rank distance codes. Des. Codes Cryptogr. 86(7), 1469–1481 (2018)

    Article  MathSciNet  Google Scholar 

  17. Stanton, D.: A partially ordered set and \(q\)-Krawtchouk polynomials. J. Comb. Theory Ser. A 30(3), 276–284 (1981)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank Ferdinand Ihringer for his feedback regarding this paper. We also thank the two anonymous reviewers whose comments helped improve and clarify this manuscript. The research of S. M. Cioabă was supported by the Grants NSF DMS-1600768, CIF-1815922, and a JSPS Invitational Fellowship for Research in Japan (Short-term S19016).

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716-2553, USA

    Sebastian M. Cioabă & Himanshu Gupta

Authors
  1. Sebastian M. Cioabă
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Himanshu Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sebastian M. Cioabă.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cioabă, S.M., Gupta, H. On the Eigenvalues of Grassmann Graphs, Bilinear Forms Graphs and Hermitian Forms Graphs. Graphs and Combinatorics 38, 30 (2022). https://doi.org/10.1007/s00373-021-02445-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00373-021-02445-z

Keywords

Search

Navigation