Journal of Algebraic Combinatorics

, Volume 24, Issue 2, pp 195–207 | Cite as

Smith’s Theorem and a characterization of the 6-cube as distance-transitive graph



A generic distance-regular graph is primitive of diameter at least two and valency at least three. We give a version of Derek Smith's famous theorem for reducing the classification of distance-regular graphs to that of primitive graphs. There are twelve cases—the generic case, four canonical imprimitive cases that reduce to the generic case, and seven exceptional cases. All distance-transitive graphs were previously known in six of the seven exceptional cases. We prove that the 6-cube is the only distance-transitive graph coming under the remaining exceptional case.


Imprimitive distance-transitive graph Imprimitive distance-regular graph 


  1. 1.
    M.R. Alfuraidan, “Imprimitive distance-transitive graphs,” Ph.D. Thesis, Michigan State University, March 2004.Google Scholar
  2. 2.
    M.R. Alfuraidan and J.I. Hall, “Imprimitive distance-transitive graphs with primitive core of diameter at least three,” in preparation.Google Scholar
  3. 3.
    N.L. Biggs and A. Gardiner, “The classification of distance transitive graphs,” unpublished manuscript, 1974.Google Scholar
  4. 4.
    J.T.M. van Bon and A.E. Brouwer, “The distance-regular antipodal covers of classical distance-regular graphs,” in: Colloq. Math. Soc. Janos Bolyai, Proc. Eger 1987, 1988, pp. 141–166.Google Scholar
  5. 5.
    A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer, Berlin, 1989.MATHGoogle Scholar
  6. 6.
    P.J. Cameron, Permutation Groups, London Mathematical Society Student Texts, vol. 45, Cambridge University Press, Cambridge, 1999.Google Scholar
  7. 7.
    C.D. Godsil, R.A. Liebler, and C.E. “Praeger, Antipodal distance transitive covers of complete graphs,” Europ. J. Combin. 19 (1998), 455–478.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J. Hemmeter, “Halved graphs, Johnson and Hamming graphs,” Utilitas Math. 25 (1984), 115–118.MATHMathSciNetGoogle Scholar
  9. 9.
    J. Hemmeter, “Distance-regular graphs and halved graphs,” Europ. J. Combin. 7 (1986), 119–129.MATHMathSciNetGoogle Scholar
  10. 10.
    A.A. Ivanov, “Distance-transitive graphs and their classification,” in Investigations in the Algebraic Theory of Combinatorial Objects, I.A. Faradzev, A.A. Ivanov, M.H. Klin, and A.J. Woldar (Eds.), Kluwer, Dordrecht, 1994, pp. 283–378.Google Scholar
  11. 11.
    A.A. Ivanov, R.A. Liebler, T. Penttila, and C.E. Praeger, “Antipodal distance-transitive covers of complete bipartite graphs,” European J. Combin. 18 (1997), 11–33.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    W.M. Kantor, “Classification of 2-transitive symmetric designs,” Graphs Combin. 1 (1985), 165–166.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    M.W. Liebeck, “The affine permutation groups of rank three,” Proc. London Math. Soc. 54(3), (1987), 477–516.Google Scholar
  14. 14.
    D.H. Smith, “Primitive and imprimitive graphs, Quart,” J. Math. Oxford 22(2), (1971), 551–557.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of Mathematical SciencesKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia
  2. 2.Department of MathematicsMichigan State UniversityEast LansingUSA

Personalised recommendations