Bounds for the trace norm of A_ matrix of digraphs

Let D be a digraph of order n with adjacency matrix A (D). For [0, 1), the A_ matrix of D is defined as A_ (D) = ^+ (D) + (1-) A (D), where ^+ (D) =diag~ (d₁^+, d₂^+, , dₙ^+) is the diagonal matrix of vertex outdegrees of D. Let ₁ (D), ₂ (D), , ₍ (D) be the singular values of A_ (D). Then the trace norm of A_ (D), which we call trace norm of D, is defined as \|A_ (D) \|_*=₈=₁^n₈ (D). In this paper, we find the singular values of some basic digraphs and characterize the digraphs D with Rank~ (A_ (D) ) =1. As an application of these results, we obtain a lower bound for the trace norm of A_ matrix of digraphs and determine the extremal digraphs. In particular, we determine the oriented trees for which the trace norm of A_ matrix attains minimum. We obtain a lower bound for the spectral norm ₁ (D) of digraphs and characterize the extremal digraphs. As an application of this result, we obtain an upper bound for the trace norm of digraphs and characterize the extremal digraphs.