  Relatively general techniques for computing mean first-passage time (MFPT) of
random walks on networks with a specific property are very useful, since a
universal method for calculating MFPT on general graphs is not available
because of their complexity and diversity. In this paper, we present techniques
for explicitly determining the partial mean first-passage time (PMFPT), i.e.,
the average of MFPTs to a given target averaged over all possible starting
positions, and the entire mean first-passage time (EMFPT), which is the average
of MFPTs over all pairs of nodes on regular treelike fractals. We describe the
processes with a family of regular fractals with treelike structure. The
proposed fractals include the $T$ fractal and the Peano basin fractal as their
special cases. We provide a formula for MFPT between two directly connected
nodes in general trees on the basis of which we derive an exact expression for
PMFPT to the central node in the fractals. Moreover, we give a technique for
calculating EMFPT, which is based on the relationship between characteristic
polynomials of the fractals at different generations and avoids the computation
of eigenvalues of the characteristic polynomials. Making use of the proposed
methods, we obtain analytically the closed-form solutions to PMFPT and EMFPT on
the fractals and show how they scale with the number of nodes. In addition, to
exhibit the generality of our methods, we also apply them to the Vicsek
fractals and the iterative scale-free fractal tree and recover the results
previously obtained.
