Merging the A- and Q-spectral theories for digraphs

Let $G$ be a digraph and $A(G)$ be the adjacency matrix of $G$. Let $D(G)$ be the diagonal matrix with outdegrees of vertices of $G$. For any real $\alpha \in [0,1]$, Liu et al. \cite{LWCL} defined the matrix $A_{\alpha }(G)$ as $$A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)\mathrm{.}$$ The largest modulus of the eigenvalues of $A_{\alpha }(G)$ 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 discuss a number of open problems.