Comparison on the spectral radii of weighted digraphs that differ in a certain subdigraph

Let DS be a weighted digraph of order n with a subdigraph S of order k, M(DS) its adjacency weight matrix and ρ(DS) its spectral radius. We consider the class Ck of weighted digraphs of order k and we study the preorder in Ck given by DS′≾DS if and only if ρ(DS′)≤ρ(DS). We obtain that this order is equivalent to the entry-wise order M(DS′)≤M(DS). Several points of view are taken, under varying regularity conditions, and k polynomial conditions for the comparison are presented.