Delsarte Set Graphs with Small c2

Let Γ be a Delsarte set graph with an intersection number c2 (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 ψ1 > 1 then c2 ≥ 2 ψ1 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 c2 = 2ψ1 > 2. As a consequence of this result, we show that if c2 ≤ 5 and ψ1 > 1 then Γ is either a Johnson graph or a folded Johnson graph $${\overline{J}(4s,2s)}$$ with s ≥ 3.