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

The trace norm of M ∈ M n ( C ) is defined as ‖ M ‖ ⁎ = ∑ k = 1 n σ k , where σ 1 ≥ σ 2 ≥ ⋯ ≥ σ n ≥ 0 are the singular values of M (i.e. the square roots of the eigenvalues of M M ⁎ ). We are particularly interested in the trace norm ‖ L ( D ) − a n I n ‖ ⁎ , where L ( D ) is the Laplacian matrix of a digraph D with n vertices and a arcs, and I n is the n × n identity matrix. When D = G is a graph with n vertices and m edges, then ‖ L ( D ) − a n I n ‖ ⁎ = ‖ L ( G ) − 2 m n I n ‖ ⁎ = L E ( 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 ) − a n I n ‖ ⁎ ≤ n ( a − a 2 n + ∑ i = 1 n ( d i + ) 2 ) , where d 1 + , … , d n + 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. [ABSTRACT FROM AUTHOR]