  We study the probability flux on the central vertex in continuous-time
quantum walks on weighted tree graphs. In a weighted graph, each edge has a
weight we call hopping. This hopping sets the jump rate of the particle between
the vertices connected by the edge. Here, the edges of the central vertex
(root) have a hopping parameter $J$ larger than those of the other edges. For
star graphs, this hopping gives only how often the walker visits the central
vertex over time. However, for weighted spider graphs $S_{n,2}$ and $S_{n,3}$,
the probability on the central vertex drops with $J^2$ for walks starting from
a state of any superposition of leaf vertices. We map Cayley trees $C_{3,2}$
and $C_{3,3}$ into these spider graphs and observe the same dependency. Our
results suggest this is a general feature of such walks on weighted trees and a
way of probing decoherence effects in an open quantum system context.
