On the Spectral Radius of (0,1)-Matrices with 1’s in Prescribed Positions

Abstract

Let $n$ and $d$ be positive integers with $1\leq d\leq n(n-1)/2$. We investigate the maximum and minimum spectral radii of a $(0,1)$-matrix of order $n$ that has $1$'s on and below its main diagonal and $d$ additional 1's. If $d\leq 4$ we determine all matrices of this type that have the maximum spectral radius. For general $d$ we prove an asymptotic result that severely limits the structure of matrices with maximum spectral radius. For $d\leq n$, we determine the minimum spectral radius.