The spread of the spectrum of a nonnegative matrix with a zero diagonal element

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Let A = [aᵢ ⱼ]ᵢ,ⱼ₌₁ⁿ be a nonnegative matrix with a₁ ₁ = 0. We prove some lower bounds for the spread s(A) of A that is defined as the maximum distance between any two eigenvalues of A. If A has only two distinct eigenvalues, then s(A) ≥ n/2(n-1) r(A), where r(A) is the spectral radius of A. Moreover, this lower bound is the best possible.