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

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.