Convergence of a Transformation on a Weighted Graph

Kramer and Bruckner defined the following transformation on a weighted graph on n vertices. The transformation replaces the weight at every vertex by either the minimum or the maximum of the weights in its closed neighborhood, the choice being that extremum which is closer in value to the original weight. They showed that the system always converges after a finite number of iterations of the transform. We consider here the question of how fast this convergence is, and show that O(n) iterations suffice. We conjecture though that n − 2 iterations suffice, and verify this for some special graphs. The convergence rates of more general transforms are also considered.