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 $A_{\alpha }(D)=\alpha {\mathrm{\Delta }}^{+}(D)+(1-\alpha )A(D)$, where ${\mathrm{\Delta }}^{+}(D)=diag(d_1^{+},d_2^{+},\dots ,d_n^{+})$ 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 $\mathrm{\parallel }{A}_{\alpha }(D){\mathrm{\parallel }}_{\ast }={\sum }_{i=1}^n{\sigma }_{i\alpha }(D)$. In this paper, we find the singular values of some basic digraphs and characterize the digraphs $D$ with $Rank(A_{\alpha }(D))=1$. 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.