Bounds for the trace norm of Aα matrix of digraphs

Abstract

Let D be a digraph of order n with adjacency matrix $A\left(D\right)$. For $\alpha \in [0,1)$, the $A_{\alpha }$ matrix of D is defined as $A_{\alpha }\left(D\right)=\alpha {\Delta }^{+}\left(D\right)+(1-\alpha )A\left(D\right)$, where ${\Delta }^{+}\left(D\right)=\text{diag}(d_1^{+},d_2^{+},\dots ,d_n^{+})$ is the diagonal matrix of vertex out degrees of D. Let ${\sigma }_{1\alpha }\left(D\right),{\sigma }_{2\alpha }\left(D\right),\dots ,{\sigma }_{n\alpha }\left(D\right)$ be the singular values of $A_{\alpha }\left(D\right)$. Then the trace norm of $A_{\alpha }\left(D\right)$, which we call α trace norm of D, is defined as ${\Vert A_{\alpha }\left(D\right)\Vert }_{⁎}={\sum }_{i=1}^n{\sigma }_{i\alpha }\left(D\right)$. In this paper, we find the singular values of some basic digraphs and characterize the digraphs D with $\text{Rank}\left(A_{\alpha }\right(D\left)\right)=1$. As an application of these results, we obtain a lower bound for the trace norm of $A_{\alpha }$ matrix of digraphs and determine the extremal digraphs. In particular, we determine the oriented trees for which the trace norm of $A_{\alpha }$ matrix attains minimum. We obtain a lower bound for the α spectral norm ${\sigma }_{1\alpha }\left(D\right)$ 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.