  We study the set of probability distributions visited by a continuous-time
quantum walk on graphs. An edge-weighted graph G is universal mixing if the
instantaneous or average probability distribution of the quantum walk on G
ranges over all probability distributions on the vertices as the weights are
varied over non-negative reals. The graph is uniform mixing if it visits the
uniform distribution. Our results include the following:
  (a) All weighted complete multipartite graphs are instantaneous universal
mixing. This is in contrast to the fact that no unweighted complete
multipartite graphs are uniform mixing (except for the four-cycle).
  (b) The weighted claw or star graph is a minimally connected instantaneous
universal mixing graph. In fact, as a corollary, the unweighted claw is
instantaneous uniform mixing. This adds a new family of uniform mixing graphs
to a list that so far contains only the hypercubes.
  (c) Any weighted graph is average almost-uniform mixing unless its spectral
type is sublinear in the size of the graph. This provides a nearly tight
characterization for average uniform mixing on circulant graphs.
  (d) No weighted graphs are average universal mixing. This shows that weights
do not help to achieve average universal mixing, unlike the instantaneous case.
  Our proofs exploit the spectra of the underlying weighted graphs and path
collapsing arguments.
