Skip to main content
SpringerLink
Log in

On distance-regular graphs withk i =k j , II

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

Abstract

LetГ be a distance-regular graph of diameterd andi-th valencyk i. We show that ifk 2 = kj for 2 +j ≥ d and 2 <j, thenГ is a polygon (k = 2) or an antipodal 2-cover (k d = 1). We also give a short proof of Terwilliger's inequality for bipartite distance-regular graphs and a refinement of Ivanov's argument on diameter bound.

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

Access this article

Log in via an institution

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

References

  1. Bannai, E., Ito, T.: Algebraic Combinatorics I: Association Schemes, Benjamin-Cummings Lecture Note Ser.58, Benjamin/Cummings, London (1984)

    Google Scholar 

  2. Bannai, E., Ito, T.: Current researches on algebraic combinatorics. Graphs and Comb.2, 287–308 (1986)

    Google Scholar 

  3. Bannai, E., Ito, T.: On distance-regular graphs with fixed valency. Graphs and Comb.3, 95–109 (1987)

    Google Scholar 

  4. Bannai, E., Ito, T.: On distance-regular graphs with fixed valency, III. J. Algebra107, 43–52 (1987)

    Google Scholar 

  5. Bannai, E., Ito, T.: On distance-regular graphs with fixed valency, IV. European J. Comb.10, 137–148 (1989)

    Google Scholar 

  6. Boshier, A., Nomura, K.: A remark on the intersection arrays of distance-regular graphs, J. Comb. Theory Ser. B44, 147–153 (1988)

    Google Scholar 

  7. Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs, Springer-Verlag, Berlin, Heidelberg, 1989

    Google Scholar 

  8. Ivanov, A.A.: Bounding the diameter of a distance-regular graph. Soviet Math. Doklady28, 149–152 (1983)

    Google Scholar 

  9. Suzuki, H.: On a distance-regular graph withb e = 1. SUT Journal of Math.29, 1–14 (1993)

    Google Scholar 

  10. Suzuki, H.: On distance-regular graphs withk i = kj. J. Comb. Theory Ser.B 61, 103–110 (1994)

    Google Scholar 

  11. Terwilliger, P.: Distance-regular graphs and (s, c, a, k)-graphs. J. Comb. Theory Ser.B 34, 156–164 (1983)

    Google Scholar 

  12. Terwilliger, P.: Distance-regular graphs with girth 3 or 4, I. J. Comb. Theory Ser.B 39, 265–281 (1985)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Osaka University, 1-1 Machikaneyama, 560, Toyonaka, Osaka, Japan

    Akira Hiraki

  2. Department of Mathematics, International Christian University, 10-2, Osawa 3-chome, Mitaka, 181, Tokyo, Japan

    Hiroshi Suzuki

  3. Department of Mathematics, Hokkaido Institute of Technology, 7-15-4-1 Maeda, Teine, 006, Sapporo, Japan

    Masayuki Wajima

Authors
  1. Akira Hiraki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hiroshi Suzuki
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Masayuki Wajima
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hiraki, A., Suzuki, H. & Wajima, M. On distance-regular graphs withk i =k j , II. Graphs and Combinatorics 11, 305–317 (1995). https://doi.org/10.1007/BF01787811

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01787811

Keywords

Search

Navigation