Let $G$ be a simple graph with adjacency matrix $A(G)$, signless Laplacian matrix $Q(G)$, degree diagonal matrix $D(G)$ and let $l(G)$ be the line graph of $G$. In 2017, Nikiforov defined the $A_\alpha$-matrix of $G$, $A_\alpha(G)$, as a linear convex combination of $A(G)$ and $D(G)$, the following way, $A_\alpha(G):=\alpha A(G)+(1-\alpha)D(G),$ where $\alpha\in[0,1]$. In this paper, we present some bounds for the eigenvalues of $A_\alpha(G)$ and for the largest and smallest eigenvalues of $A_\alpha(l(G))$. Extremal graphs attaining some of these bounds are characterized.

翻译：设$G$是一个简单图，其邻接矩阵为$A(G)$，无符号拉普拉斯矩阵为$Q(G)$，度对角矩阵为$D(G)$，并记$l(G)$为$G$的线图。2017年，Nikiforov定义了$G$的$A_\alpha$-矩阵$A_\alpha(G)$，作为$A(G)$和$D(G)$的线性凸组合，具体形式为$A_\alpha(G):=\alpha A(G)+(1-\alpha)D(G)$，其中$\alpha\in[0,1]$。在本文中，我们给出了$A_\alpha(G)$的特征值以及$A_\alpha(l(G))$的最大和最小特征值的若干界。同时，确定了达到某些界的极图。