Some New Lower Bounds for the Spread of a Nonnegative Matrix with a Zero Diagonal Element

Let Nn (with n ≥ 2) be the family of all nonnegative n × n matrices A = [aij ], where a11 = 0 and the remaining entries aij ∈ [0, 1) with a spectral radius ρ(A) = 1. We can establish a lower bound for the additional spread s(A) ≥ k/ n−1 , where k is the count of zero diagonal elements in matrix A. Furthermore, if matrix A possesses only two distinct eigenvalues, then it follows that s(A) ≥ n−2/n−1 . Additionally, we derived a few other lower bounds under a special family of matrices.