Using symbolic computation to prove nonexistence of distance-regular graphs

A package for the Sage computer algebra system is developed for checking feasibility of a given intersection array for a distance-regular graph. We use this tool to show that there is no distance-regular graph with intersection array {(2r + 1)(4r + 1)(4t− 1), 8r(4rt− r + 2t), (r + t)(4r + 1); 1, (r + t)(4r + 1), 4r(2r + 1)(4t− 1)} (r, t ≥ 1), {135,128,16; 1,16,120}, {234,165,12; 1,30,198} or {55,54,50,35,10; 1,5,20,45,55}. In all cases, the proofs rely on the equality in the Krein condition, from which triple intersection numbers are determined. Further combinatorial arguments are then used to derive nonexistence.