Generalizations of Bounds on the Index of Convergence to Weighted Digraphs

TL;DR

This paper extends classical bounds on the index of convergence in unweighted digraphs to weighted digraphs, focusing on optimal walks and their periodicity in max-algebraic matrix powers, with implications for strongly connected graphs.

Contribution

It generalizes known bounds on the index of convergence to weighted digraphs, particularly when one end of optimal walks is a critical node.

Findings

Bounds of Wielandt, Dulmage-Mendelsohn, Schwarz, Kim, and Gregory-Kirkland-Pullman apply to weighted digraphs with critical nodes.

The transient of periodicity depends on graph size and weight magnitude.

Results connect classical unweighted bounds to weighted cases in max-algebraic settings.

Abstract

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.