For an Erdős-Rényi graph with fixed edge probability, the random graph Laplacian has simple spectrum with overwhelmingly high probability, with a quantitative n^{-3/2-o(1)} lower bound on the minimum gap.
Extreme gaps between eigenvalues of random matrices.Ann
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Eigenvalue gaps of the Laplacian of random graphs
For an Erdős-Rényi graph with fixed edge probability, the random graph Laplacian has simple spectrum with overwhelmingly high probability, with a quantitative n^{-3/2-o(1)} lower bound on the minimum gap.