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 a 3 . We will show that a 3 divides k for a Shilla distance-regular graph Γ , and for Γ we define b = b ( Γ ) ≔ k a 3 . 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.