On the Perron roots of principal submatrices of co-order one of irreducible nonnegative matrices (Q1863561)

This paper is devoted to the relationships between the spectral properties of a nonnegative irreducible matrix \(A\) and those of its principal submatrices of co-order 1. Let \(\lambda(A)\) be the Perron root (spectral radius) of \(A\) and \(\lambda_{\min}(A)\) the minimum of the Perron roots of all the principal submatrices of \(A\) of co-order 1. It is well known that the interval \((\lambda_{\min} (A)\), \(\lambda(A))\) does not contain any eigenvalues of \(A\). What can one say about the spectral properties of the point \(\lambda_{ \min} (A)\) as an eigenvalue of \(A\)? When is this point the second largest real eigenvalue of \(A\)? In this paper the author completely reduces these problems to the determination of the spectral properties of the Perron root of a principal submatrix of \(A\) of co-order 1 with spectral radius \(\lambda_{\min} (A)\).