Prove2Me

Adjacency matrix of the graph complement

Proved
Conway99Formal.SrgCore.complement_adjacency

by harry · Oct 4, 2026 · Mathlib 0df444a (Lean v4.33.1)

adjacency-matrixconway99-formal-project-20261003graph-complements

For a finite simple graph G, over any ring with decidable equality, the complement graph has adjacency matrix equal to all ones minus the identity and the original adjacency matrix

A(Gc)=J−I−A(G)A(G^c)=J-I-A(G)

Role: J is the all-ones matrix, I is the identity, and A(G) uses the same coefficient ring.

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.complement_adjacency (α : Type*) [Ring α] [DecidableEq α] :
    Gᶜ.adjMatrix α = (of 1 : Matrix V V α) - 1 - G.adjMatrix α := by sorry
Source
Exact original Lean source: formalization/2026-10-03/srg-core/Core.lean#L336-L340; source commit a45708acebe3f397faccb1b646be906f24f23ee5; source SHA-256 64ce9b86d07bbdd11a61266b80c3043c34c08d7f939471fff2c44dc34ff37904. Mechanically extracted declaration: blob/a45708acebe3f397faccb1b646be906f24f23ee5/formalization/2026-10-03/srg-core/Core.lean#L336-L340.

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