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Delsarte Set Graphs with Small c 2

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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.

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Correspondence to S. Bang.

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Bang, S., Hiraki, A. & Koolen, J.H. Delsarte Set Graphs with Small c 2 . Graphs and Combinatorics 26, 147–162 (2010).

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