The Tutte polynomial of a graph, or equivalently the q-state Potts model partition

function, is a two-variable polynomial graph invariant of considerable importance in

both combinatorics and statistical physics. The computation of this invariant for

a graph is NP-hard in general. The aim of this paper is to compute the Tutte

polynomial of the Apollonian network. Based on the well-known duality property of

the Tutte polynomial, we extend the subgraph-decomposition method. In particular,

we do not calculate the Tutte polynomial of the Apollonian network directly, instead

we calculate the Tutte polynomial of the Apollonian dual graph. By using the close

relation between the Apollonian dual graph and the Hanoi graph, we express the

Tutte polynomial of the Apollonian dual graph in terms of that of the Hanoi graph.

As an application, we also give the number of spanning trees of the Apollonian

network.