Let G be a digraph and $A\left(G\right)$ be the adjacency matrix of G. Let $D\left(G\right)$ be the diagonal matrix with outdegrees of vertices of G. For any real $\alpha \in [0,1]$, define the matrix $A_{\alpha }\left(G\right)$ as $A_{\alpha }\left(G\right)=\alpha D\left(G\right)+(1-\alpha )A\left(G\right).$ The largest modulus of the eigenvalues of $A_{\alpha }\left(G\right)$ is called the $A_{\alpha }$ spectral radius of G. In this paper, we determine the digraphs which attain the maximum (or minimum) $A_{\alpha }$ spectral radius among all strongly connected digraphs with given parameters such as girth, clique number, vertex connectivity or arc connectivity. We also propose an open problem.

中文翻译：

---

给定参数的有向图的Aα谱半径

设G是一个有向图并且$\mathrm{一个}\left(G\right)$是G的邻接矩阵。让$D\left(G\right)$是G的顶点出度的对角矩阵。对于任何真实的$\alpha \in [0,1]$, 定义矩阵${\mathrm{一个}}_{\alpha }\left(G\right)$作为${\mathrm{一个}}_{\alpha }\left(G\right)=\alpha D\left(G\right)+(1-\alpha )\mathrm{一个}\left(G\right).$的特征值的最大模数${\mathrm{一个}}_{\alpha }\left(G\right)$被称为${\mathrm{一个}}_{\alpha }$G的光谱半径。在本文中，我们确定达到最大值（或最小值）的有向图${\mathrm{一个}}_{\alpha }$具有给定参数（例如周长、团数、顶点连通性或弧连通性）的所有强连通有向图之间的谱半径。我们还提出了一个未解决的问题。