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

Let A=[aij]i,j=1n be a nonnegative matrix with a11=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)⩾n2(n−1)r(A), where r(A) is the spectral radius of A. Moreover, this lower bound is the best possible.