math.CO — Combinatorics
Nonexistence of DSRGs with ten parameter sets: the rank of A − θI versus a submatrix prescribed at a vertex
For a directed strongly regular graph (DSRG), the parameters determine rank(A − θI) at nontrivial eigenvalues θ. For small λ or t = λ + 1 they almost determine a piece of this matrix at a vertex: a piece needing more rank excludes the graph, and a small margin leaves finitely many cases. We exclude (50, 11, 7, 0, 3), (68, 13, 7, 0, 3), (100, 16, 7, 0, 3), (108, 19, 9, 0, 4), (48, 9, 5, 0, 2), (55, 16, 12, 1, 6), (66, 10, 5, 4, 1) and (91, 12, 6, 5, 1), open in Hobart and Brouwer's table, and their complements. Counting twin classes reproves (96, 15, 10, 9, 1) and (28, 6, 3, 2, 1): for the first, rank replaces one counting bound; for the second, it is unused. Lean 4 with Mathlib proves nine exclusions; for (28, 6, 3, 2, 1) its kernel also checks a SAT certificate. (48, 9, 5, 0, 2) ends with a search by two separately written programs, the second certified by CaDiCaL unsatisfiability proofs checked by drat-trim.