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Graphs and Combinatorics

, Volume 26, Issue 2, pp 147–162 | Cite as

Delsarte Set Graphs with Small c 2

  • S. Bang
  • A. Hiraki
  • J. H. Koolen
Original Paper

Abstract

Let Γ be a Delsarte set graph with an intersection number c 2 (i.e., a distance-regular graph with a set \({\mathcal{C}}\) of Delsarte cliques such that each edge lies in a positive constant number \({n_{\mathcal{C}}}\) of Delsarte cliques in \({\mathcal{C}}\)). We showed in Bang et al. (J Combin 28:501–506, 2007) that if ψ 1 > 1 then c 2 ≥ 2 ψ 1 where \({\psi_1:=|\Gamma_1(x)\cap C |}\) for \({x\in V(\Gamma)}\) and C a Delsarte clique satisfying d(x, C) = 1. In this paper, we classify Γ with the case c 2 = 2ψ 1 > 2. As a consequence of this result, we show that if c 2 ≤ 5 and ψ 1 > 1 then Γ is either a Johnson graph or a folded Johnson graph \({\overline{J}(4s,2s)}\) with s ≥ 3.

Keywords

Delsarte clique Delsarte set graph Distance-regular graph Johnson graph 

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References

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    Bang S., Hiraki A., Koolen J.H.: Delsarte clique graphs. Eur. J. Combin. 28, 501–516 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
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    Brouwer A.E., Cohen A.M., Neumaier A.: Distance-regular Graphs. Springer, Berlin (1989)zbMATHGoogle Scholar
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    Godsil C.D.: Algebraic Combinatorics. Chapman and Hall, New York (1993)zbMATHGoogle Scholar
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    van Lint J.H., Wilson R.M.: A Course in Combinatorics. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar

Copyright information

© Springer 2010

Authors and Affiliations

  1. 1.Department of MathematicsPusan National UniversityPusanRepublic of Korea
  2. 2.Division of Mathematical SciencesOsaka Kyoiku UniversityKashiwaraJapan
  3. 3.Department of Mathematics, Pohang Mathematics InstitutePOSTECHNamgu, PohangRepublic of Korea

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