Spielman-Srivastava spectral sparsification preserves Laplacian quadratic forms to within (1 +/- epsilon), but does not directly control the adjacency spectral radius lambda_1, which governs the NIMFA epidemic threshold and arises in spectral clustering. We prove |lambda_1(A_H) - lambda_1(A_G)| <= epsilon(2 Delta - lambda_1) deterministically, with a sharp epsilon*lambda_1 bound for reweighting sparsifiers via Perron-Frobenius monotonicity. Under effective-resistance sampling, Matrix Bernstein gives O(epsilon Delta / sqrt(c)) with high probability. Combining eigenvector delocalization with resolvent perturbation theory, we establish that for graphs with delocalized Perron eigenvectors and spectral gap = Omega(Delta), the distortion is O(epsilon Delta sqrt(log n) / sqrt(n)) + O(epsilon^2 Delta^2 / delta_gap), with corollaries for Erdos-Renyi graphs, regular expanders, and stochastic block models. Lower bounds establish tightness for regular graphs.

翻译：Spielman-Srivastava谱稀疏化将拉普拉斯二次型保持在(1 ± ε)误差范围内，但并未直接控制邻接谱半径λ₁——该参数主导NIMFA流行病阈值并出现在谱聚类中。我们确定性证明|λ₁(A_H) - λ₁(A_G)| ≤ ε(2Δ - λ₁)，并借助Perron-Frobenius单调性对重加权稀疏化得到尖锐的ε*λ₁界。在有效电阻采样下，Matrix Bernstein方法以高概率给出O(εΔ/√c)的界值。结合特征向量离域性与预解摄动理论，我们建立：对于具有离域Perron特征向量及谱间隙Ω(Δ)的图，其畸变上界为O(εΔ√(log n)/√n) + O(ε²Δ²/δ_gap)，并得到对于Erdős-Rényi图、正则扩张图及随机块模型的推论。下界验证了正则图的紧致性。