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

Abstract

Let $D_S$ be a weighted digraph of order n with a subdigraph S of order k, $M(D_S)$ its adjacency weight matrix and $\rho (D_S)$ its spectral radius. We consider the class $C_k$ of weighted digraphs of order k and we study the preorder in $C_k$ given by $D_{S^{\prime }}\precsim D_S$ if and only if $\rho (D_{S^{\prime }})\le \rho (D_S)$. We obtain that this order is equivalent to the entry-wise order $M(D_{S^{\prime }})\le M(D_S)$. Several points of view are taken, under varying regularity conditions, and k polynomial conditions for the comparison are presented.