  Efficiently controlling the diffusion process is crucial in the study of
diffusion problem in complex systems. In the sense of random walks with a
single trap, mean trapping time(MTT) and mean diffusing time(MDT) are good
measures of trapping efficiency and diffusion efficiency respectively. They
both vary with the location of the node. In this paper, we study random walks
on T-fractal and provided general methods to calculate the MTT for any target
node and the MDT for any source node. Using the MTT and the MDT as the measure
of trapping efficiency and diffusion efficiency respectively, we compare the
trapping efficiency and diffusion efficiency among all nodes of T-fractal and
find the best (or worst) trapping sites and the best (or worst) diffusing
sites. Our results show that: the hub node of T-fractal is the best trapping
site, but it is also the worst diffusing site, the three boundary nodes are the
worst trapping sites, but they are also the best diffusing sites. Comparing the
minimum and maximum of MTT and MDT, we found that the maximum of MTT is almost
$6$ times of the minimum for MTT and the maximum of MDT is almost equal to the
minimum for MDT. These results show that the location of target node has big
effect on the trapping efficiency, but the location of source node almost has
no effect on diffusion efficiency. We also conducted numerical simulation to
test the results we have derived, the results we derived are consistent with
those obtained by numerical simulation.
