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Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean

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@@ -195,9 +195,10 @@ more often found in the literature, where `J` is the all-ones matrix. -/
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theorem IsSRGWith.matrix_eq {α : Type*} [Semiring α] (h : G.IsSRGWith n k ℓ μ) :
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G.adjMatrix α ^ 2 = k • (1 : Matrix V V α) + ℓ • G.adjMatrix α + μ • Gᶜ.adjMatrix α := by
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ext v w
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simp only [adjMatrix_pow_apply_eq_card_walk, Set.coe_setOf, Matrix.add_apply, Matrix.smul_apply,
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simp only [adjMatrix_pow_apply_eq_card_walk, Matrix.add_apply, Matrix.smul_apply,
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adjMatrix_apply, compl_adj]
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erw [Fintype.card_congr (G.walkLengthTwoEquivCommonNeighbors v w)]
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rw [@Fintype.card_congr _ _ (G.fintypeSetWalkLength v w 2) _
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(G.walkLengthTwoEquivCommonNeighbors v w)]
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obtain rfl | hn := eq_or_ne v w
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· rw [← Set.toFinset_card]
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simp [commonNeighbors, ← neighborFinset_def, h.regular v]

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