Relation between the trace norm of an oriented graph and its rank

Let D be an oriented graph, i.e., a digraph in which all arcs are not symmetric, with vertex set {1, ... , n}. The adjacency matrix A = (aij) of D is the n x n matrix defined as aij = 1 if ij is an arc of D and aij = 0 if ij is not an arc of D. The rank of D, written as r(D), is defined to be the rank of its adjacency matrix, and the trace norm of D is defined asN(D) = n i=1 & sigma;i, where & sigma;1 > & sigma;2 >... >& sigma;n > 0 are the singular values of A, i.e., the square roots of the eigenvalues of AAT. In this paper, we establish a lower bound and an upper bound for the trace norm of a connected oriented graph D in terms of its rank and its maximum vertex degree & UDelta; as r(D) < N(D) < r(D)& UDelta;, the extremal graphs attaining the bounds are characterized, -& RARR; then the lower bound can be improved to r(D) +.V5 -2, where respectively. Furthermore, we prove that if D is not Pnor Cn-& RARR;, Pn and -& RARR; -& RARR; Cn respectively denote the orientation of Pn and Cn such that all 2-degree vertices are not sink-source. & COPY; 2023 Elsevier Inc. All rights reserved.