Summary: For each \(n \geq 2\), let \(A_{n } = (\xi _{ij})\) be an \(n \times n\) symmetric matrix with diagonal entries equal to zero and the entries in the upper triangular part being independent with mean \(\mu _{n}\) and standard deviation \(\sigma _{n}\). The Laplacian matrix is defined by \(\Delta_n=\) diag\((\sum_{j=1}^n\xi_{ij})_{1\leq i \leq n}-A_n\). In this paper, we obtain the laws of large numbers for \(\lambda _{n-k }(\Delta _{n })\), the \((k + 1)\)-th smallest eigenvalue of \(\Delta _{n}\), through the study of the order statistics of weakly dependent random variables. Under certain moment conditions on \(\xi _{ij}\)’s, we prove that, as \(n \rightarrow \infty\),

(i)

\(\frac{\lambda_{n-k}(\Delta_n)-n\mu_n} {\sigma_n\sqrt{n\log n}} \to -\sqrt{2}\) a.s. for any \(k \geq 1\).

Further, if \({\Delta _{n }; n \geq 2}\) are independent with \(\mu _{n } = 0\) and \(\sigma _{n } = 1\), then,

(ii)

the sequence \({\;\left\{\frac{\lambda_{n-k}(\Delta_n)}{\sqrt{n\log n}};n\geq 2\right\}}\) is dense in \({\left[-\sqrt{2+2(k+1)^{-1}}, -\sqrt{2}\,\right] \text{ a.s.}}\) for any \(k \geq 0\).

In particular, (i) holds for the Erdös-Rényi random graphs. Similar results are also obtained for the largest eigenvalues of \(\Delta _{n }\).