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  "path": "/abs/2606.07459v1",
  "publishedAt": "2026-06-08T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Joshua Steier"
  ],
  "textContent": "**Authors:** Joshua Steier\n\nSpielman-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.",
  "title": "Adjacency Spectral Radius Under Laplacian Sparsification: Deterministic and Probabilistic Bounds"
}