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Journal of Algebraic Combinatorics

, Volume 43, Issue 4, pp 771–782 | Cite as

Asymptotic Delsarte cliques in distance-regular graphs

  • László BabaiEmail author
  • John Wilmes
Article

Abstract

We give a new bound on the parameter \(\lambda \) (number of common neighbors of a pair of adjacent vertices) in a distance-regular graph G, improving and generalizing bounds for strongly regular graphs by Spielman (1996) and Pyber (2014. arXiv:1409.3041). The new bound is one of the ingredients of recent progress on the complexity of testing isomorphism of strongly regular graphs (Babai et al. 2013). The proof is based on a clique geometry found by Metsch (Des Codes Cryptogr 1(2):99–116, 1991) under certain constraints on the parameters. We also give a simplified proof of the following asymptotic consequence of Metsch’s result: If \(k\mu = o(\lambda ^2)\), then each edge of G belongs to a unique maximal clique of size asymptotically equal to \(\lambda \), and all other cliques have size \(o(\lambda )\). Here k denotes the degree and \(\mu \) the number of common neighbors of a pair of vertices at distance 2. We point out that Metsch’s cliques are “asymptotically Delsarte” when \(k\mu = o(\lambda ^2)\), so families of distance-regular graphs with parameters satisfying \(k\mu = o(\lambda ^2)\) are “asymptotically Delsarte-geometric.”

Keywords

Distance-regular graphs Clique Clique geometry Delsarte clique Asymptotic analysis 

Notes

Acknowledgments

The authors wish to acknowledge the inspiration from their joint work with Xi Chen, Xiaorui Sun, and Shang-Hua Teng on the isomorphism problem for strongly regular graphs.

References

  1. 1.
    Babai, L\’{a}szl\’{o}: On the complexity of canonical labeling of strongly regular graphs. SIAM J. Comput. 9(1), 212–216 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Babai, László, Chen, Xi, Sun, Xiaorui, Teng, Shang-Hua, Wilmes, John: Faster canonical forms for strongly regular graphs. In: Proc. of the 54th Annual Symp. on Foundations of Computer Science (FOCS’13), pp. 157–166. IEEE Computer Society (2013)Google Scholar
  3. 3.
    Babai, László, Kantor, William M., Luks, Eugene M.: Computational complexity and the classification of finite simple groups. In: Proc. 24th Annual Symp. on Foundations of Computer Science (FOCS’83), pp. 162–171. IEEE Computer Society (1983)Google Scholar
  4. 4.
    Babai, László, Luks, Eugene M.: Canonical labeling of graphs. In: Proc. 15th ACM Symp. on Theory of Computing (STOC’83), pp. 171–183. ACM Press (1983)Google Scholar
  5. 5.
    Brouwer, Andries E.: Parameters of strongly regular graphs. http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html
  6. 6.
    Brouwer, Andries E., Cohen, Arjeh M., Neumaier, Arnold: Distance-Regular Graphs. Springer, Berlin (1989)Google Scholar
  7. 7.
    Brouwer, Andries E., Neumaier, Arnold: A remark on partial linear spaces of girth 5 with an application to strongly regular graphs. Combinatorica 8(1), 57–61 (1988)Google Scholar
  8. 8.
    Delsarte, Philippe: An algebraic approach to the association schemes of coding theory. Ph.D. Thesis. Philips Res. Rep. Suppl. no. 10, pp. vi+97 (1973)Google Scholar
  9. 9.
    Godsil, Christopher D.: Algebraic Combinatorics. Chapman & Hall, New York (1993)Google Scholar
  10. 10.
    Godsil, Christopher D.: Geometric distance-regular covers. New Zealand J. Math. 22(2), 31–38 (1993)Google Scholar
  11. 11.
    Koolen, Jack H., Bang, Sejeong: On distance-regular graphs with smallest eigenvalue at least \(-m\). J. Comb. Theory Ser. B 100(6), 573–584 (2010)Google Scholar
  12. 12.
    Metsch, Klaus: Improvement of Bruck’s completion theorem. Des. Codes Cryptogr. 1(2), 99–116 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Metsch, Klaus: On a characterization of bilinear forms graphs. Eur. J. Comb. 20(4), 293–306 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Neumaier, Arnold: Strongly regular graphs with smallest eigenvalue \(-m\). Arch. Math. 33(4), 392–400 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pyber, László: Large connected strongly regular graphs are Hamiltonian. arXiv:1409.3041 (2014)
  16. 16.
    Spielman, Daniel A.: Faster isomorphism testing of strongly regular graphs. In: Proc. 28th ACM Symp. on Theory of Computing (STOC’96), pp. 576–584. ACM Press (1996)Google Scholar
  17. 17.
    Sun, Xiaorui, Wilmes, John: Faster canonical forms for primitive coherent configurations. In: Proc. 47th ACM Symp. on Theory of Computing (STOC’15), pp. 693–702, ACM Press (2015)Google Scholar
  18. 18.
    Zemlyachenko, Viktor N., Korneenko, Nikolai M., Tyshkevich, Regina I.: Graph isomorphism problem. Zap. Nauchn. Sem. (LOMI) 118, 83–158, 215 (1982)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Departments of Computer Science and MathematicsUniversity of ChicagoChicagoUSA

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