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

Abstract

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×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 as N(D)=∑i=1nσi, where σ1≥σ2≥…≥σn≥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 Δ asr(D)≤N(D)≤r(D)Δ, the extremal graphs attaining the bounds are characterized, respectively. Furthermore, we prove that if D is not Pn→ or Cn→, then the lower bound can be improved to r(D)+5−2, where Pn→ and Cn→ respectively denote the orientation of Pn and Cn such that all 2-degree vertices are not sink-source.