TY - JOUR
T1 - Low eigenvalues of Laplacian matrices of large random graphs
AU - Jiang, Tiefeng
PY - 2012/8
Y1 - 2012/8
N2 - 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-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, 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.
AB - 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-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, 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.
KW - Extreme eigenvalues
KW - Laplacian matrix
KW - Order statistics
KW - Random graph
KW - Random matrix
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U2 - 10.1007/s00440-011-0357-4
DO - 10.1007/s00440-011-0357-4
M3 - Article
AN - SCOPUS:84864392145
SN - 0178-8051
VL - 153
SP - 671
EP - 690
JO - Probability Theory and Related Fields
JF - Probability Theory and Related Fields
IS - 3-4
ER -