Skip to main content
SpringerLink
Log in

Strongly Closed Subgraphs in a Distance-Regular Graph with c 2 > 1

  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

Let Γ be a distance-regular graph of diameter d ≥ 3 with c 2 > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions:

  • (SC) m : For any pair of vertices at distance m there exists a strongly closed subgraph of diameter m containing them.

  • (BB) m : Let (x, y, z) be a triple of vertices with ∂ Γ (x, y) = 1 and ∂ Γ (x, z) = ∂ Γ (y, z)  =  m. Then B(x, z) = B(y, z).

  • (CA) m : Let (x, y, z) be a triple of vertices with ∂ Γ (x, y) = 2, ∂ Γ (x, z) = ∂ Γ (y, z) = m and |C(z, x) ∩ C(z, y)| ≥ 2. Then C(x, z) ∪ A(x, z) = C(y, z) ∪ A(y, z).

Suppose that the condition (SC) m holds. Then it has been known that the condition (BB) i holds for all i with 1 ≤ i ≤ m. Similarly we can show that the condition (CA) i holds for all i with 1 ≤ i ≤ m. In this paper we prove that if the conditions (BB) i and (CA) i hold for all i with 1 ≤ i ≤ m, then the condition (SC) m holds. Applying this result we give a sufficient condition for the existence of a dual polar graph as a strongly closed subgraph in Γ.

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. Benjamin-Cummings, California, (1984)

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

  3. Cameron, P.J.: Dual polar spaces. Geom. Dedicata. 12, 75–85 (1982)

    Google Scholar 

  4. Hiraki, A.: Distance-regular subgraphs in a distance-regular graph, V. European J. Combin. 19, 141–150 (1998)

    Google Scholar 

  5. Hiraki, A.: Distance-regular subgraphs in a distance-regular graph, VI. European J. Combin. 19, 953–965 (1998)

    Google Scholar 

  6. Hiraki, A.: An application of a construction theory of strongly closed subgraphs in a distance-regular graph. European J. Combin. 20, 271–278 (1999)

    Google Scholar 

  7. Hiraki, A.: Strongly closed subgraphs in a regular thick near polygon. European J. Combin. 20, 789–796 (1999)

    Google Scholar 

  8. Hiraki, A.: A distance-regular graph with strongly closed subgraphs. J. Algebraic Combin. 14, 127–131 (2001)

    Google Scholar 

  9. Hiraki, A.: A characterization of the doubled Grassmann graphs, the doubled Odd graphs, and the Odd graphs by strongly closed subgraphs. European J. Combin. 24, 161–171 (2003)

    Google Scholar 

  10. Hiraki, A.: A distance-regular graph with bipartite geodetically closed subgraphs. European J. Combin. 24, 349–363 (2003)

    Google Scholar 

  11. Hiraki, A.: A characterization of the Hamming graph by strongly closed subgraphs. European J. Combin. 29, 1603–1616 (2008)

    Google Scholar 

  12. Hiraki, A.: A characterization of some distance-regular graphs by strongly closed subgraphs, to appear in European J. Combin

  13. Hiraki, A.: Distance-regular graph with c 2 > 1 and a 1 = 0 <  a 2, to appear in Graphs and Combin

  14. Suzuki, H.: On strongly closed subgraphs of highly regular graphs, European J. Combin. 16, 197–220 (1995)

    Google Scholar 

  15. Weng, C.-W.: D-bounded distance-regular graphs. European J. Combin. 18, 211–229 (1997)

    Google Scholar 

  16. Weng, C.-W.: Weak-geodetically closed subgraphs in distance-regular graphs. Graphs and Combin. 14, 275–304 (1998)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Division of Mathematical Sciences, Osaka Kyoiku University, Osaka, 582-8582, Japan

    Akira Hiraki

Authors
  1. Akira Hiraki
    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. Strongly Closed Subgraphs in a Distance-Regular Graph with c 2 > 1. Graphs and Combinatorics 24, 537–550 (2008). https://doi.org/10.1007/s00373-008-0814-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-008-0814-8

Keywords

Search

Navigation