A Sharp upper bound for the spectral radius of a nonnegative matrix and applications

TL;DR

This paper derives a precise upper bound for the spectral radius of nonnegative matrices and applies it to various graph spectral radii, providing new or generalized bounds for graphs and digraphs.

Contribution

It introduces a sharp upper bound for the spectral radius of nonnegative matrices and extends this to multiple spectral radii of graphs and digraphs, generalizing existing results.

Findings

Established a sharp upper bound for nonnegative matrix spectral radius

Derived new bounds for adjacency, Laplacian, and signless Laplacian spectral radii of graphs

Extended bounds to distance and distance Laplacian spectral radii

Abstract

In this paper, we obtain a sharp upper bound for the spectral radius of a nonnegative matrix. This result is used to present upper bounds for the adjacency spectral radius, the Laplacian spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance Laplacian spectral radius, the distance signless Laplacian spectral radius of a graph or a digraph. These results are new or generalize some known results.