Shilla distance-regular graphs

A Shilla distance-regular graph ΓΓ (say with valency kk) is a distance-regular graph with diameter 3 such that its second-largest eigenvalue equals a3a3. We will show that a3a3 divides kk for a Shilla distance-regular graph ΓΓ, and for ΓΓ we define b=b(Γ)≔ka3. In this paper we will show that there are finitely many Shilla distance-regular graphs ΓΓ with fixed b(Γ)≥2b(Γ)≥2. Also, we will classify Shilla distance-regular graphs with b(Γ)=2b(Γ)=2 and b(Γ)=3b(Γ)=3. Furthermore, we will give a new existence condition for distance-regular graphs, in general.