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Let $G$ be a simple connected graph of order $n$ with degree sequence $(d_1,d_2,...,d_n)$. Denote ${{(}^{\alpha }t)}_i={\sum }_{j:i\sim j}{d}_j^{\alpha }$, ${{(}^{\alpha }m)}_i={{(}^{\alpha }t)}_i/d_i^{\alpha }$ and ${{(}^{\alpha }N)}_i={\sum }_{j:i\sim j}{{(}^{\alpha }t)}_j$, where $\alpha$ is a real number. Denote by ${\lambda }_1\left(G\right)$ and ${\mu }_1\left(G\right)$ the spectral radius of the adjacency matrix and the Laplacian matrix of $G$, respectively. In this paper, we present some upper and lower bounds of ${\lambda }_1\left(G\right)$ and ${\mu }_1\left(G\right)$ in terms of ${{(}^{\alpha }t)}_i$, ${{(}^{\alpha }m)}_i$ and ${{(}^{\alpha }N)}_i$. Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results.