Shilla distance-regular graphs

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