Triple intersection numbers of -polynomial distance-regular graphs

We use the system of linear Diophantine equations introduced by Coolsaet and Jurišić to obtain information about a feasible family of distance-regular graphs with vanishing Krein parameter q 22 1 and intersection arrays { ( r + 1 ) ( r 3 − 1 ) , r ( r − 1 ) ( r 2 + r − 1 ) , r 2 − 1 ; 1 , r ( r + 1 ) , ( r 2 − 1 ) ( r 2 + r − 1 ) } , r ≥ 2 . In this way we are able to calculate certain triple intersection numbers and prove nonexistence for all r ≥ 3 . For r = 3 nonexistence was not known before, however it is well known that the intersection array for r = 2 uniquely determines the halved 7-cube. Then we show how to apply Terwilliger balanced set conditions for Q -polynomial distance-regular graphs to produce additional linear Diophantine equations.