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- W1995825863 abstract "For each n ≥ 2, let A n = (ξ ij ) be an n × n symmetric matrix with diagonal entries equal to zero and the entries in the upper triangular part being independent with mean μ n and standard deviation σ n . The Laplacian matrix is defined by $${{bf Delta}_n={rm diag}(sum_{j=1}^nxi_{ij})_{1leq i leq n}-{bf A}_n}$$ . In this paper, we obtain the laws of large numbers for λ n–k (Δ n ), the (k + 1)-th smallest eigenvalue of Δ n , through the study of the order statistics of weakly dependent random variables. Under certain moment conditions on ξ ij ’s, we prove that, as n → ∞, $$({rm i})quadfrac{lambda_{n-k}({bf Delta}_n)-nmu_n} {sigma_nsqrt{nlog n}} to -sqrt{2} quad a.s. $$ for any k ≥ 1. Further, if {Δ n ; n ≥ 2} are independent with μ n = 0 and σ n = 1, then, (ii) the sequence $${;left{frac{lambda_{n-k}({bf Delta}_n)}{sqrt{nlog n}};ngeq 2right}}$$ is dense in $${left[-sqrt{2+2(k+1)^{-1}}, -sqrt{2},right] a.s.}$$ for any k ≥ 0. In particular, (i) holds for the Erdös–Rényi random graphs. Similar results are also obtained for the largest eigenvalues of Δ n ." @default.
- W1995825863 created "2016-06-24" @default.
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- W1995825863 date "2011-03-20" @default.
- W1995825863 modified "2023-10-14" @default.
- W1995825863 title "Low eigenvalues of Laplacian matrices of large random graphs" @default.
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- W1995825863 doi "https://doi.org/10.1007/s00440-011-0357-4" @default.
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