Let $D\left(G\right)$ and $T_r\left(G\right)$ be the distance matrix and diagonal matrix with vertex transmissions of a connected graph $G,$ separately. Define matrix $D_{\alpha }\left(G\right)$ as $D_{\alpha }\left(G\right)=\alpha T_r\left(G\right)+(1-\alpha )D\left(G\right),$ $0\le \alpha \le 1$. Let $U_n=\left\{G\right|G$ is a simple connected graph with $\left|V\right(G\left)\right|=\left|E\right(G\left)\right|=n\},$ $T_n=\left\{T\right|T$ is a tree of order $n\}$ and their complement sets be $U_n^c$ and $T_n^c,$ separately. In this paper, we generalize the conclusions in Qin et al. (2020) to $D_{\alpha }$-matrix: we depict the extremal graph with maximum $D_{\alpha }$-spectral radius among $U_n^c$ ($n\ge 8$) for any $\alpha \in [0,\frac{1}{2}],$ and also characterize the graphs among $T_n^c$ that reach the maximum and minimum of $D_{\alpha }$-spectral radius for any $\alpha \in [0,1],$ respectively.

让 $d（G）$ 和 ${Ť}_{\mathrm{[R}}（G）$ 是连接图的顶点传输的距离矩阵和对角矩阵 $G，$分别。定义矩阵$d_{\alpha }（G）$ 如 $d_{\alpha }（G）=\alpha {Ť}_{\mathrm{[R}}（G）+（\mathrm{1个}-\alpha ）d（G），$ $0\le \alpha \le \mathrm{1个}$。让${ü}_{ñ}=\left\{G\right|G$ 是一个简单的连接图 $|V（G）|=|Ë（G）|=ñ\}，$ ${Ť}_{ñ}=\left\{Ť\right|Ť$ 是秩序树 $ñ\}$ 他们的补集是 ${ü}_{ñ}^C$ 和 ${Ť}_{ñ}^C，$分别。在本文中，我们概括了Qin等人的结论。（2020年）至$d_{\alpha }$-matrix：我们描绘了最大的极值图 $d_{\alpha }$光谱半径 ${ü}_{ñ}^C$ （$ñ\ge 8$） $\alpha \in [0，\frac{\mathrm{1个}}{2}]，$ 并描述其中的图形 ${Ť}_{ñ}^C$ 达到最大和最小 $d_{\alpha }$-任何光谱半径 $\alpha \in [0，\mathrm{1个}]，$ 分别。