Low eigenvalues of Laplacian matrices of large random graphs

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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. In this paper, we obtain the laws of large numbers for λ n-kn), 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, is dense in, 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.

Original languageEnglish (US)
Pages (from-to)671-690
Number of pages20
JournalProbability Theory and Related Fields
Volume153
Issue number3-4
DOIs
StatePublished - Aug 1 2012

Keywords

  • Extreme eigenvalues
  • Laplacian matrix
  • Order statistics
  • Random graph
  • Random matrix

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