  Let $G$ be a graph of order $n$ with $m$ edges. Also let $\mu_1\geq \mu_2\geq
\cdots\geq \mu_{n-1}\geq \mu_n=0$ be the Laplacian eigenvalues of graph $G$ and
let $\sigma=\sigma(G)$ $(1\leq \sigma\leq n)$ be the largest positive integer
such that $\mu_{\sigma}\geq \frac{2m}{n}$. In this paper, we prove that
$\mu_2(G)\geq \frac{2m}{n}$ for almost all graphs. Moreover, we characterize
the extremal graphs for any graphs. Finally, we provide the answer to Problem 3
in \cite{KMT}, that is, the characterization of all graphs with $\sigma=1$.
