Variation in $α$ 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 $α\in[0,1)$, the $A_α$ matrix of $D$ is defined as $A_α(D)=αΔ^{+}(D)+(1-α)A(D)$, where $Δ^{+}(D)=\mbox{diag}~(d_1^{+},d_2^{+},\dots,d_n^{+})$ is the diagonal matrix of vertex outdegrees of $D$. Let $σ_{1α}(D),σ_{2α}(D),\dots,σ_{nα}(D)$ be the singular values of $A_α(D)$. Then the trace norm of $A_α(D)$, which we call $α$ trace norm of $D$, is defined as $\|A_α(D)\|_*=\sum_{i=1}^{n}σ_{iα}(D)$. In this paper, we study the variation in $α$ 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 $α$ trace norm.