New bounds on the periodicity transient of the powers of a tropical matrix: Using cyclicity and factor rank

Building on the weak CSR approach developed in a previous paper by Merlet, Nowak and Sergeev, we establish new bounds for periodicity threshold of powers tropical matrix. According to that approach, ultimate take form T=max(T_1,T_2), where T_1 is bound time after which expansion starts hold T_2 first term dominate. The established this make use cyclicity associated graph (tropical) factor rank matrix, leads much improved favorable cases. For T_1, particular, obtain extensions Schwarz, Kim Gregory-Kirkland-Pullman, previously known as exponents digraphs. similar T_2, introduce novel concept walk reduction it both rank.