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.
Similar content being viewed by others
References
Bang S., Hiraki A., Koolen J.H.: Delsarte clique graphs. Eur. J. Combin. 28, 501–516 (2007)
Brouwer A.E., Cohen A.M., Neumaier A.: Distance-regular Graphs. Springer, Berlin (1989)
Godsil C.D.: Algebraic Combinatorics. Chapman and Hall, New York (1993)
van Lint J.H., Wilson R.M.: A Course in Combinatorics. Cambridge University Press, Cambridge (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bang, S., Hiraki, A. & Koolen, J.H. Delsarte Set Graphs with Small c 2 . Graphs and Combinatorics 26, 147–162 (2010). https://doi.org/10.1007/s00373-010-0905-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-010-0905-1