Bounds for the trace norm of Aα matrix of digraphs

Let D be a digraph of order n with adjacency matrix $A(D)$. For α∈[0,1), the Aα matrix of D is defined as Aα(D)=α Δ⁺(D)+(1-α)A(D), where Δ⁺(D)=diag~(d₁⁺,d₂⁺,,dₙ⁺) is the diagonal matrix of vertex outdegrees of D. Let σ1α(D),σ2α(D),,σnα(D) be the singular values of Aα(D). Then the trace norm of Aα(D), which we call α trace norm of D, is defined as \|Aα(D)\|_*=∑ᵢ₌₁ⁿσiα(D). In this paper, we find the singular values of some basic digraphs and characterize the digraphs D with Rank~(Aα(D))=1. As an application of these results, we obtain a lower bound for the trace norm of Aα matrix of digraphs and determine the extremal digraphs. In particular, we determine the oriented trees for which the trace norm of Aα matrix attains minimum. We obtain a lower bound for the α spectral norm σ1α(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.