Average path length for Sierpinski pentagon

In this paper,we investigate diameter and average path length(APL) of Sierpinski pentagon based on its recursive construction and self-similar structure.We find that the diameter of Sierpinski pentagon is just the shortest path lengths between two nodes of generation 0. Deriving and solving the linear homogenous recurrence relation the diameter satisfies, we obtain rigorous solution for the diameter. We also obtain approximate solution for APL of Sierpinski pentagon, both diameter and APL grow approximately as a power-law function of network order $N(t)$, with the exponent equals $\frac{\ln(1+\sqrt{3})}{\ln(5)}$. Although the solution for APL is approximate,it is trusted because we have calculated all items of APL accurately except for the compensation($\Delta_{t}$) of total distances between non-adjacent branches($\Lambda_t^{1,3}$), which is obtained approximately by least-squares curve fitting. The compensation($\Delta_{t}$) is only a small part of total distances between non-adjacent branches($\Lambda_t^{1,3}$) and has little effect on APL. Further more,using the data obtained by iteration to test the fitting results, we find the relative error for $\Delta_{t}$ is less than $10^{-7}$, hence the approximate solution for average path length is almost accurate.