Variation in $\alpha$ trace norm of a digraph by deletion of a vertex or an arc and its applications

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 study the variation in $\alpha$ trace norm of a digraph when a vertex or an arc is deleted. As an application of these results, we characterize oriented trees and unicyclic digraphs with maximum $\alpha$ trace norm.