Adjacency and common-neighbor row identity
ProvedConway99Formal.SrgCore.far_pair_common_neighbor_rowadjacency-matrixcommon-neighborsconway99-formal-project-20261003strongly-regular-graphs
For any distinct vertices u and v in a graph satisfying the strongly regular parameters (99,14,1,2), the integer sum over w of A(u,w)A(v,w), plus A(u,v), is 2, where A is the integer adjacency matrix. The sum is therefore 1 for adjacent pairs and 2 for nonadjacent pairs.
Preamble
import Mathlib
namespace Conway99Formal.SrgCore
end Conway99Formal.SrgCore
set_option autoImplicit false
/-! Graph-owned parameter and adjacency identities for a hypothetical SRG(99,14,1,2).
Sources: `Conway99/Conway99/Core.lean` §§1–3, 8.1;
`Conway99/Conway99/Claims/C01srgcorealgebra.lean` §§0, 3, 6;
`Conway99/results/R005_star_complement_square_discriminant.md`.
-/
open Conway99Formal.SrgCore
open SimpleGraph Matrix Finset
variable {V : Type*} [Fintype V] [DecidableEq V]
variable (G : SimpleGraph V) [DecidableRel G.Adj]
Formal statement
theorem Conway99Formal.SrgCore.far_pair_common_neighbor_row (h : G.IsSRGWith 99 14 1 2)
(u v : V) (huv : u ≠ v) :
(∑ w, G.adjMatrix ℤ u w * G.adjMatrix ℤ v w) + G.adjMatrix ℤ u v = 2 := by sorry
Source
Exact original Lean source: formalization/2026-10-03/srg-core/Core.lean#L401-L413; source commit a45708acebe3f397faccb1b646be906f24f23ee5; source SHA-256 64ce9b86d07bbd11a61266b80c3043c34c08d7f939471fff2c44dc34ff37904. Mechanically extracted declaration: blob/a45708acebe3f397faccb1b646be906f24f23ee5/formalization/2026-10-03/srg-core/Core.lean#L401-L413.