Proof that toughness t(Γ) ≥ μ₂ / (μₙ − δ) for connected graphs Γ with Laplacian eigenvalues 0 = μ₁ < μ₂ ≤ ⋯ ≤ μₙ and minimum degree δ.
Gu,A proof of Brouwer’s toughness conjecture, SIAM J
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A proof of Haemers' toughness conjecture
Proof that toughness t(Γ) ≥ μ₂ / (μₙ − δ) for connected graphs Γ with Laplacian eigenvalues 0 = μ₁ < μ₂ ≤ ⋯ ≤ μₙ and minimum degree δ.