Using symbolic computation to prove nonexistence of distance-regular graphs

Abstract

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 (Formula Presented), {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 equality in the Krein condition, from which triple intersection numbers are determined. Further combinatorial arguments are then used to derive nonexistence.