Let $G= (V, E)$ be a non-regular connected graph whose degree is bounded. Suppose that each node contain a unique token.
I want to uniformly shuffle the tokens amongst the graph using only local swaps (i.e. exchange of the tokens between two adjacent nodes) ? Is there a lower bound known for this problem ?
The only idea I had is to use a random walk result, then to see how much swaps I need to "simulate" the effect of random walks transporting tokens on the graph.