Generalizations of bounds on the index of convergence to weighted digraphs

Sequences of maximum-weight walks of a growing length in weighted digraphs have many applications in manufacturing and transportation systems, as they encode important performance parameters. It is well-known that they eventually enter a periodic regime if the digraph is strongly connected. The length of their transient phase depends, in general, both on the size of digraph and on the magnitude of the weights. In this paper, we show that certain bounds on the transients of unweighted digraphs, such as the bounds of Wielandt, Dulmage-Mendelsohn, Schwarz, Kim and Gregory-Kirkland-Pullman, remain true for critical nodes in weighted digraphs.