A note on triangle-free distance-regular graphs with

Let Γ denote a distance-regular graph with classical parameters ( D , b , α , β ) and D ⩾ 3 . Assume the intersection numbers a 1 = 0 and a 2 ≠ 0 . We show that the intersection number c 2 is either 1 or 2, and if c 2 = 1 , then ( b , α , β ) = ( − 2 , − 2 , ( ( − 2 ) D + 1 − 1 ) / 3 ) .