Enumeration of spanning trees on Apollonian networks

In this paper we study the problem of enumerating spanning trees on Apollonian networks and obtain an exact expression for the number of spanning trees, which is relevant to some dynamical characteristics of networks, such as reliability, synchronization and random walks. In addition, we compare the entropy of spanning trees on our networks with the other fractal networks, and find that the entropy of our networks is the highest among the studied fractal networks. (paper)