Shilla distance-regular graphs

A Shilla distance-regular graph Gamma (say with valency k) is a distance-regular graph with diameter 3 such that as second-largest eigenvalue equals a(3). We will show that a(3) divides k for a Shilla distance-regular graph Gamma, and for Gamma we define b = b(Gamma) = k/a(3) In this paper we will show that there are finitely many Shilla distance-regular graphs Gamma with fixed b(Gamma) >= 2 Also, we will classify Shilla distance-regular graphs with b(Gamma) = 2 and b(Gamma) = 3 Furthermore, we will give a new existence condition for distance-regular graphs, in general. (C) 2010 Elsevier Ltd. All rights reserved