On the Spectral Radius of Complementary Acyclic Matrices of Zeros and Ones

Abstract

For an $n \times n$ complementary acyclic matrix A of 0’s and l’s we show that the spectral radius $\rho ( A )$ of A satisfies $\rho ( A )\geqq n - 2$ and determine those matrices A for which equality holds. When A is an $n \times n$ irreducible, complementary tree matrix, we also obtain that $\rho ( A )\leqq \rho _n $, where $\rho _n $ is the largest root of the polynomial $\lambda^3 - (n - 2 )\lambda^2 - ( n - 3 )\lambda - 1$.