Upper bound for the trace norm of the Laplacian matrix of a digraph and normally regular digraphs

The trace norm of M∈Mn(C) is defined as ‖M‖⁎=∑k=1nσk, where σ1≥σ2≥⋯≥σn≥0 are the singular values of M (i.e. the square roots of the eigenvalues of MM⁎). We are particularly interested in the trace norm ‖L(D)−anIn‖⁎, where L(D) is the Laplacian matrix of a digraph D with n vertices and a arcs, and In is the n×n identity matrix. When D=G is a graph with n vertices and m edges, then ‖L(D)−anIn‖⁎=‖L(G)−2mnIn‖⁎=LE(G), the Laplacian energy of G introduced by Gutman and Zhou in 2006. We show that for a digraph D with n vertices and a arcs,‖L(D)−anIn‖⁎≤n(a−a2n+∑i=1n(di+)2), where d1+,…,dn+ are the outer degrees of the vertices of D. Moreover, the digraphs where this bound is attained are special classes of normally regular digraphs studied by Jørgensen in 2015 [6]. Finally, we construct normally regular digraphs where the equality is attained.