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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₍, ₂ and S₍, ₃, the probability on the central vertex drops with J² for walks starting from a state of any superposition of leaf vertices. We map Cayley trees C₃, ₂ and C₃, 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.
Vieira et al. (Fri,) studied this question.