Shilla distance-regular graphs

A Shilla distance-regular graph Γ (say with valency k) is a distance-regular graph with diameter 3 such that its second-largest eigenvalue equals a3. We will show that a3 divides k for a Shilla distance-regular graph Γ and for Γ we define b=b(Γ):;ka3. In this paper we will show that there are finitely many Shilla distance-regular graphs Γ with fixed b(Γ)≥;2. Also, we will classify Shilla distance-regular graphs with b(Γ)=2 and b(Γ)=3. Furthermore, we will give a new existence condition for distance-regular graphs, in general.