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}{74,54,15;1,9,60} and of distance-regular graphs with intersection arrays{4r3+8r2+6r+1,2r(r+1)(2r+1),2r2+2r+1;1,2r(r+1),(2r+1)(2r2+2r+1)}{4r3+8r2+6r+1,2r(r+1)(2r+1),2r2+2r+1;1,2r(r+1),(2r+1)(2r2+2r+1)} with r   an integer and r⩾1r⩾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.