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 \({{\bf \Delta}_n={\rm diag}(\sum_{j=1}^n\xi_{ij})_{1\leq i \leq n}-{\bf A}_n}\). 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 → ∞,
for any k ≥ 1. Further, if {Δ n ; n ≥ 2} are independent with μ n = 0 and σ n = 1, then, (ii) the sequence \({\;\left\{\frac{\lambda_{n-k}({\bf \Delta}_n)}{\sqrt{n\log n}};n\geq 2\right\}}\) 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 .
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T. Jiang was supported in part by NSF #DMS-0449365.
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Jiang, T. Low eigenvalues of Laplacian matrices of large random graphs. Probab. Theory Relat. Fields 153, 671–690 (2012). https://doi.org/10.1007/s00440-011-0357-4
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DOI: https://doi.org/10.1007/s00440-011-0357-4
Keywords
- Random graph
- Random matrix
- Laplacian matrix
- Extreme eigenvalues
- Order statistics
Mathematics Subject Classification (2000)
- 05C80
- 05C50
- 15A52
- 60B10
- 62G30