Bounds for the trace norm of $A_α$ matrix of digraphs

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 find the singular values of some basic digraphs and characterize the digraphs $D$ with $\mbox{Rank}~(A_α(D))=1$. As an application of these results, we obtain a lower bound for the trace norm of $A_α$ matrix of digraphs and determine the extremal digraphs. In particular, we determine the oriented trees for which the trace norm of $A_α$ matrix attains minimum. We obtain a lower bound for the $α$ spectral norm $σ_{1α}(D)$ of digraphs and characterize the extremal digraphs. As an application of this result, we obtain an upper bound for the $α$ trace norm of digraphs and characterize the extremal digraphs.