Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8897802 | Linear Algebra and its Applications | 2018 | 16 Pages |
Abstract
The trace norm of MâMn(C) is defined as âMââ=âk=1nÏk, where Ï1â¥Ï2â¥â¯â¥Ïnâ¥0 are the singular values of M (i.e. the square roots of the eigenvalues of MMâ). We are particularly interested in the trace norm âL(D)âanInââ, where L(D) is the Laplacian matrix of a digraph D with n vertices and a arcs, and In is the nÃn identity matrix. When D=G is a graph with n vertices and m edges, then âL(D)âanInââ=âL(G)â2mnInââ=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,âL(D)âanInâââ¤n(aâa2n+âi=1n(di+)2), where d1+,â¦,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.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Natalia Agudelo, Juan Rada, Mauricio Rivera,