An Approach to Bounding the Spectral Radius of a Weighted Digraph

Abstract

The paper suggests a general approach to deriving upper bounds for the spectral radii of weighted digraphs. The approach is based on the generalized Wielandt lemma (GWL), which reduces the problem of bounding the spectral radius of a given block matrix to bounding the Perron root of the matrix composed of the norms of the blocks. In the case of the adjacency matrix of weighted graphs and digraphs where all the blocks are square positive (semi)definite matrices of the same order, the GWL takes an especially nice simple form. The second component of the approach consists in applying known upper bounds for the Perron root of a nonnegative matrix. It is shown that the approach suggested covers, in particular, the known upper bounds of the spectral radius and allows one to describe the equality cases.