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We study the transients of matrices in max-plus algebra. Our approach is based on the weak CSR expansion. Using this expansion, the transient can be expressed by max{ T1 ,  T2 }, where  T1  is the weak CSR threshold and  T2  is the time after which the purely pseudoperiodic CSR terms start to dominate in the expansion. Various bounds have been derived for  T1  and  T2 , naturally leading to the question which matrices, if any, attain these bounds. In the present paper, we characterize the matrices attaining two particular bounds on  T1 , which are generalizations of the bounds of Wielandt and Dulmage–Mendelsohn on the indices of non-weighted digraphs. This also leads to a characterization of tightness for the same bounds on the transients of critical rows and columns. The characterizations themselves are generalizations of those for the non-weighted case.

On the tightness of bounds for transients of weak CSR expansions and periodicity transients of critical rows and columns of tropical matrix powers