In [27] we show that any distance-regular graph Г containing a cycle {ν 0, ν 1, ν 2, ν 3, ν 0} with δ (ν 0, ν 2)= δ (ν 1, ν 3)= 2 was finite, with diameter d, valency k and intersection numbers a 1, c d satisfying d⩽ k+ c d a i+ 2 with equality holding if and only if (1) c i− c i− 1+ b i− 1− b i− a 1− 2= 0,(1⩽ i⩽ d) In this graph we give a simplified proof of this fact, and then classify the graphs where the diameter bound is attained. Not assuming the existence of the above cycle in Γ but only that c 2> 2, we use the classification of the classical root systems to show a distance-regular graph satisfying (1) is either the Johnson graph J (d, n), a graph with the intersection numbers of the Hamming graph H (d, q), the Cocktail Party graph CP (n), ½ H (n, 2)(the halved graph of the n-cube), or one of a finite number of exceptional graphs, all with d≤ 8, a 1≤ 16, and k≤ 28.