设$G_\omega=(G, \omega)$是一个赋权图, 其邻接矩阵和赋权度对角矩阵分别$A(G_\omega)$和$D(G_\omega)$。对于$\alpha\in[0, 1]$, $G_\omega$的$A_\alpha$-矩阵为$ A_\alpha(G_\omega)=\alpha D(G_\omega)+(1-\alpha)A(G_\omega)$。对于连通赋权非正则图$G_\omega$, 给出了其关于$A_\alpha$-特征值的一些界, 并得到了$A_\alpha$-谱半径对应的特征向量中最大分量与最小分量比值的下界。

Let $G_\omega=(G, \omega)$ be a weighted graph, whose adjacency matrix and weighted degree diagnoal matrix are $A(G_\omega)$ and $D(G_\omega)$, respectively. For given $\alpha \in [0, 1]$, the matrix $A_\alpha(G_\omega)=\alpha D(G_\omega)+(1-\alpha)A(G_\omega)$ is the $A_\alpha$- matrix of $G_\omega$. In this paper, we give some bounds on the $A_\alpha$-eigenvalue of connected weighted non-regular graphs $G_\omega$, and obtain the lower bound of the ratio of the largest component to the smallest component in the eigenvector of the $A_\alpha$- spectral radius.