Using equality in the Krein conditions to prove nonexistence of certain distance-regular graphs

We prove the nonexistence of a distance-regular graph with intersection array { 74 , 54 , 15 ; 1 , 9 , 60 } and of distance-regular graphs with intersection arrays { 4 r 3 + 8 r 2 + 6 r + 1 , 2 r ( r + 1 ) ( 2 r + 1 ) , 2 r 2 + 2 r + 1 ; 1 , 2 r ( r + 1 ) , ( 2 r + 1 ) ( 2 r 2 + 2 r + 1 ) } with r an integer and r ⩾ 1 . Both cases serve to illustrate a technique which can help in determining structural properties for distance-regular graphs and association schemes with a sufficient number of vanishing Krein parameters.