# On the Tightness of Bounds for Transients of Weak CSR Expansions and Periodicity Transients of Critical Rows and Columns of Tropical Matrix Powers

## TL;DR

This paper investigates the bounds on transients in max-plus algebra matrices, characterizing matrices that attain these bounds and extending classical graph theory bounds to the weighted case.

## Contribution

It characterizes matrices that attain specific bounds on the weak CSR threshold, generalizing Wielandt and Dulmage-Mendelsohn bounds to weighted digraphs in max-plus algebra.

## Key findings

- Characterization of matrices attaining bounds on $T_1$
- Extension of classical bounds to weighted digraphs
- Insights into periodicity transients of critical matrix components

## Abstract

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 $\max\{T_1,T_2\}$, where $T_1$ is the weak CSR threshold and $T_2$ is the time after which the purely pseudoperiodic CSR terms start to dominate in 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 particular bounds on $T_1$, 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.