  We establish the first nontrivial lower bound on the (higher) topological
complexity of the unordered configuration spaces of a general graph. As an
application, we show that, for most graphs, the topological complexity
eventually stabilizes at its maximal possible value, a direct analogue of a
stability phenomenon in the ordered setting first conjectured by Farber. We
estimate the stable range in terms of the number of trivalent vertices.
