Tutte polynomial of the Apollonian network

Abstract

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, in general, NP-hard. 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.