A matrix realization of spectral bounds

TL;DR

This paper introduces a systematic method using modified quotient matrices to bound the largest eigenvalue of nonnegative matrices and identifies matrices with maximum eigenvalues among (0,1)-matrices, solving a longstanding problem.

Contribution

It presents a unified approach for spectral bounds and characterizes matrices with maximal eigenvalues among (0,1)-matrices, resolving a problem posed in 1985.

Findings

Unified method for spectral bounds using quotient matrices

Characterization of matrices with maximum eigenvalues among (0,1)-matrices

Resolution of a longstanding open problem from 1985

Abstract

We give a unified and systematic way to find bounds for the largest real eigenvalue of a nonnegative matrix by considering its modified quotient matrix. We leverage this insight to identify the unique class of matrices whose largest real eigenvalue is maximum among all $(0,1)$-matrices with a specified number of ones. This result resolves a problem that was posed independently by R. Brualdi and A. Hoffman, as well as F. Friedland, back in 1985.