Bounds for the trace norm of $A_{\alpha}$ matrix of digraphs

Let $D$ be a digraph of order $n$ with adjacency matrix $A(D)$. For $\alpha\in[0,1)$, the $A_{\alpha}$ matrix of $D$ is defined as $A_{\alpha}(D)=\alpha {\Delta}^{+}(D)+(1-\alpha)A(D)$, where ${\Delta}^{+}(D)=\mbox{diag}~(d_1^{+},d_2^{+},\dots,d_n^{+})$ is the diagonal matrix of vertex outdegrees of $D$. Let $\sigma_{1\alpha}(D),\sigma_{2\alpha}(D),\dots,\sigma_{n\alpha}(D)$ be the singular values of $A_{\alpha}(D)$. Then the trace norm of $A_{\alpha}(D)$, which we call $\alpha$ trace norm of $D$, is defined as $\|A_{\alpha}(D)\|_*=\sum_{i=1}^{n}\sigma_{i\alpha}(D)$. In this paper, we find the singular values of some basic digraphs and characterize the digraphs $D$ with $\mbox{Rank}~(A_{\alpha}(D))=1$. As an application of these results, we obtain a lower bound for the trace norm of $A_{\alpha}$ matrix of digraphs and determine the extremal digraphs. In particular, we determine the oriented trees for which the trace norm of $A_{\alpha}$ matrix attains minimum. We obtain a lower bound for the $\alpha$ spectral norm $\sigma_{1\alpha}(D)$ of digraphs and characterize the extremal digraphs. As an application of this result, we obtain an upper bound for the $\alpha$ trace norm of digraphs and characterize the extremal digraphs.