Potts model partition functions on two families of fractal lattices

• We study two families of fractal lattices with self-similar structures.
• Their Potts model partition functions are obtained by subgraph decomposition method.
• Their spanning tree numbers and asymptotic growth constants are determined.

The partition function of qq-state Potts model, or equivalently the Tutte polynomial, is computationally intractable for regular lattices. The purpose of this paper is to compute partition functions of qq-state Potts model on two families of fractal lattices. Based on their self-similar structures and by applying the subgraph-decomposition method, we divide their Tutte polynomials into two summands, and for each summand we obtain a recursive formula involving the other summand. As a result, the number of spanning trees and their asymptotic growth constants, and a lower bound of the number of connected spanning subgraphs or acyclic root-connected orientations for each of such two lattices are obtained.