Graphs and Combinatorics

, Volume 26, Issue 5, pp 629–646 | Cite as

Intriguing Sets of Vertices of Regular Graphs

  • Bart De BruynEmail author
  • Hiroshi Suzuki
Original Paper


Intriguing and tight sets of vertices of point-line geometries have recently been studied in the literature. In this paper, we indicate a more general framework for dealing with these notions. Indeed, we show that some of the results obtained earlier can be generalized to larger classes of graphs. We also give some connections and relations with other notions and results from algebraic graph theory. One of the main tools in our study will be the Bose–Mesner algebra associated with the graph.


Regular graph Tight set Intriguing set Completely regular code T-Design 

Mathematics Subject Classification (2000)

05E30 15A42 


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  1. 1.
    Bamberg J., De Clerck F., Durante N.: A hemisystem of a nonclassical generalised quadrangle. Des. Codes Cryptogr. 51, 157–165 (2009)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bamberg, J., De Clerck, F., Durante, N.: Intriguing sets of partial quadrangles. Preprint, arXiv:0812.2871Google Scholar
  3. 3.
    Bamberg, J., Giudici, M., Royle, G.F.: Every flock generalised quadrangle has a hemisystem. Preprint, arXiv:0912.2574Google Scholar
  4. 4.
    Bamberg J., Kelly S., Law M., Penttila T.: Tight sets and m-ovoids of finite polar spaces. J. Combin. Theory Ser. A 114, 1293–1314 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bamberg J., Law M., Penttila T.: Tight sets and m-ovoids of generalised quadrangles. Combinatorica 29, 1–17 (2009)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Brouwer A.E., Cohen A.M., Neumaier A.: Distance-regular Graphs. Springer, Berlin (1989)zbMATHGoogle Scholar
  7. 7.
    Brouwer A.E., Godsil C.D., Koolen J.H., Martin W.J.: Width and dual width of subsets in polynomial association schemes. J. Combin. Theory Ser. A 102, 255–271 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cossidente A., Penttila T.: Hemisystems on the Hermitian surface. J. Lond. Math. Soc. (2) 72, 731–741 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cossidente, A., Penttila, T.: A new hemisystem of H(3,49). Ars Combin. (to appear)Google Scholar
  10. 10.
    De Bruyn B.: Tight sets of points in the half-spin geometry related to Q +(9, q). Linear Algebra Appl. 424, 480–491 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    De Bruyn B.: A characterization of m-ovoids and i-tight sets of polar spaces. Adv. Geom. 8, 367–375 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Delsarte, P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. No. 10, 97 pp. (1973)Google Scholar
  13. 13.
    Godsil C.: Algebraic Combinatorics. Chapman and Hall Mathematics Series. Chapman & Hall, New York (1993)Google Scholar
  14. 14.
    Godsil C., Royle G.: Algebraic graph theory. Graduate Texts in Mathematics, vol. 207. Springer, New York (2001)Google Scholar
  15. 15.
    Haemers, W.H.: Eigenvalue techniques in design and graph theory. Ph.D. thesis, Technical University Eindhoven (1979)Google Scholar
  16. 16.
    Haemers W.H.: Interlacing eigenvalues and graphs. Linear Algebra Appl. 226/228, 593–616 (1995)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Meyerowitz A.: Cycle-balance conditions for distance-regular graphs. Discrete Math. 264, 149–165 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Neumaier, A.: Regular sets and quasisymmetric 2-designs. Combinatorial theory (Schloss Rauischholzhausen, 1982), pp. 258–275. Lecture Notes in Mathematics, vol. 969, Springer, Berlin (1982)Google Scholar
  19. 19.
    Neumaier A.: Completely regular codes. Discrete Math. 106/107, 353–360 (1992)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Payne S.E.: Tight pointsets in finite generalized quadrangles. Congr. Numer. 60, 243–260 (1987)MathSciNetGoogle Scholar
  21. 21.
    Payne S.E.: Tight pointsets in finite generalized quadrangles, II. Congr. Numer. 77, 31–41 (1990)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Read R.C., Wilson R.J.: An atlas of graphs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1998)Google Scholar
  23. 23.
    Tanaka H.: Classification of subsets with minimal width and dual width in Grassmann, bilinear forms and dual polar graphs. J. Combin. Theory Ser. A 113, 903–910 (2006)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 2010

Authors and Affiliations

  1. 1.Department of Pure Mathematics and Computer AlgebraGhent UniversityGhentBelgium
  2. 2.Department of Mathematics and Computer ScienceInternational Christian UniversityTokyoJapan

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