Remarks on spectral radius and Laplacian eigenvalues of a graph

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