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

Abstract

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 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.