  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
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 then we compare with those of other graphs with
the same average degree.
