Delsarte Set Graphs with Small c ₂

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.