Distance-regular graphs of diameter 3 having eigenvalue −1.

For a distance-regular graph of diameter three Γ, the statement that distance-3 graph Γ 3 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 a 3 = α ( b 1 c 2 = β and a 3 ≠ 0 , respectively). Such distance-regular graphs for small numbers α = 1 , 2 or β = 3 with a 3 ≠ 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. [ABSTRACT FROM AUTHOR]