Distance-regular graphs of diameter 3 having eigenvalue −1

For a distance-regular graph of diameter three Γ, the statement that distance-3 graph Γ3 of Γ is strongly regular is equivalent to that Γ has eigenvalue −1. There are many distance-regular graphs of diameter 3 having eigenvalue −1, such as the folded 7-cube, generalized hexagons of order (s,s) and antipodal nonbipartite distance-regular graphs of diameter 3. In this paper, we show that for a fixed positive integer α (β, respectively), there are only finitely many distance-regular graphs of diameter 3 having eigenvalue −1 and a3=α (b1c2=β and a3≠0, respectively). Such distance-regular graphs for small numbers α=1,2 or β=3 with a3≠0 are classified. We show that there are no distance-regular graphs with intersection array {44,35,3;1,5,42}. Moreover, we classify the distance-regular graphs with diameter 3 and smallest eigenvalue greater than −3.