The Number of Spanning Trees in Apollonian Networks

In this paper we find an exact analytical expression for the number of spanning trees in Apollonian networks. This parameter can be related to significant topological and dynamic properties of the networks, including percolation, epidemic spreading, synchronization, and random walks. As Apollonian networks constitute an interesting family of maximal planar graphs which are simultaneously small-world, scale-free, Euclidean and space filling, modular and highly clustered, the study of their spanning trees is of particular relevance. Our results allow also the calculation of the spanning tree entropy of Apollonian networks, which we compare with those of other graphs with the same average degree.