Using symbolic computation to prove nonexistence of distance-regular graphs

Janoš Vidali

paper

Using symbolic computation to prove nonexistence of distance-regular graphs

arXiv:1803.10797

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 ${(2 r + 1) (4 r + 1) (4 t - 1), 8 r (4 r t - r + 2 t), (r + t) (4 r + 1); 1, (r + t) (4 r + 1), 4 r (2 r + 1) (4 t - 1)}$ ( $r, t \geq 1$ ), ${135, 128, 16; 1, 16, 120}$ , ${234, 165, 12; 1, 30, 198}$ or ${55, 54, 50, 35, 10; 1, 5, 20, 45, 55}$ . In all cases, the proofs rely on equality in the Krein condition, from which triple intersection numbers are determined. Further combinatorial arguments are then used to derive nonexistence.