  In this paper, we study the trapping problem in two representative polymer
networks, Cayley trees and Vicsek fractals, which separately model dendrimers
and regular hyperbranched polymers. Our goal is to explore the impact of trap
location on the efficiency of trapping in these two important polymer systems,
with the efficiency being measured by the average trapping time (ATT) that is
the average of source-to-trap mean first-passage time over every staring point
in the whole networks. For Cayley trees, we derive an exact analytic formula
for the ATT to an arbitrary trap node, based on which we further obtain the
explicit expression of ATT for the case that the trap is uniformly distributed.
For Vicsek fractals, we provide the closed-form solution for ATT to a
peripheral node farthest from the central node, as well as the numerical
solutions for the case when the trap is placed on other nodes. Moreover, we
derive the exact formula for the ATT corresponding to the trapping problem when
the trap has an uniform distribution over all nodes. Our results show that the
influence of trap location on the trapping efficiency is completely different
for the two polymer networks. In Cayley trees, the leading scaling of ATT
increases with the shortest distance between the trap and the central node,
implying that trap's position has an essential impact on the trapping
efficiency; while in Vicsek fractals, the effect of location of the trap is
negligible, since the dominant behavior of ATT is identical, respective of the
location where the trap is placed. We also present that for all cases of
trapping problems being studied, the trapping process is more efficient in
Cayley trees than in Vicsek fractals. We demonstrate that all differences
related to trapping in the two polymer systems are rooted in their underlying
topological structures.
