Mean trapping time for an arbitrary trap site on a class of fractal scale-free trees

Fractals are ubiquitous in nature and random walks on fractals have attracted lots of scientific attention in the past several years. In this work, we consider discrete random walks on a class of fractal scale-free trees (FST), whose topologies are controlled by two integer parameters (i.e., u≥2 and v≥1) and exhibit a wide range of topological properties by suitably varying the parameters u and v. The mean trapping time (MTT), referred to as T_{y}, which is the mean time it takes the walker to be absorbed by the trap fixed at site y of the FST, is addressed analytically on the FST, and the effects of the trap location y on the MTT for the FST and for the general trees are also analyzed. First, a method, which is based on the connection between the MTT and the effective resistances, to derive analytically T_{y} for an arbitrary site y of the FST is presented, and some examples are provided to show the effectiveness of the method. Then, we compare T_{y} for all the possible site y of the trees, and find the sites where T_{y} achieves the minimum (or maximum) on the FST. Finally, we analyze the effects of trap location on the MTT in general trees and find that the average path length (APL) from an arbitrary site to the trap is the decisive factor which dominates the difference in the MTTs for different trap locations on general trees. We find, for any tree, the MTT obtains the minimum (or maximum) at sites where the APL achieves the minimum (or maximum).