  We study sequences of optimal walks of a growing length, in weighted
digraphs, or equivalently, sequences of entries of max-algebraic matrix powers
with growing exponents. It is known that these sequences are eventually
periodic when the digraphs are strongly connected. The transient of such
periodicity depends, in general, both on the size of digraph and on the
magnitude of the weights. In this paper, we show that some bounds on the
indices of periodicity of (unweighted) digraphs, such as the bounds of
Wielandt, Dulmage-Mendelsohn, Schwarz, Kim and Gregory-Kirkland-Pullman, apply
to the weights of optimal walks when one of their ends is a critical node.
