Let $G$ be a graph with $n$ vertices, $m$ edges and a vertex degree sequence $(d_1,d_2,\cdots ,d_n)$, where $d_1\ge d_2\ge \cdots \ge d_n$. The spectral radius and the largest Laplacian eigenvalue are denoted by $\rho \left(G\right)$ and $\mu \left(G\right)$, respectively. We determine the graphs with $$\rho \left(G\right)=\frac{d_n-1}{2}+\sqrt{2m-n{d}_n+\frac{{(d_n+1)}^2}{4}}$$ and the graphs with $d_n\ge 1$ and $$\mu \left(G\right)=d_n+\frac{1}{2}+\sqrt{{\sum }_{i=1}^n{d}_i(d_i-d_n)+{\left(d_n-\frac{1}{2}\right)}^2}.$$ We also present some sharp lower bounds for the Laplacian eigenvalues of a connected graph.