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We study the transients of matrices in max-plus algebra. Our approach isbased on the weak CSR expansion. Using this expansion, the transient can beexpressed by $\max\{T_1,T_2\}$, where $T_1$ is the weak CSR threshold and $T_2$is the time after which the purely pseudoperiodic CSR terms start to dominatein the expansion. Various bounds have been derived for $T_1$ and $T_2$,naturally leading to the question which matrices, if any, attain these bounds.  In the present paper we characterize the matrices attaining two particularbounds on $T_1$, which are generalizations of the bounds of Wielandt andDulmage-Mendelsohn on the indices of non-weighted digraphs. This also leads toa characterization of tightness for the same bounds on the transients ofcritical rows and columns. The characterizations themselves are generalizationsof 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