Using symbolic computation to prove nonexistence of distance-regular graphs
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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.
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