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 \({\Delta }^{+}(D)=\text{ diag }~(d_1^{+},d_2^{+},\dots ,d_n^{+})\) 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 \(\Vert A_{\alpha }(D)\Vert _*=\sum _{i=1}^{n}\sigma _{i\alpha }(D)\) . 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.
