Exact Potts/Tutte Polynomials for Hammock Chain Graphs

Abstract

We present exact calculations of the q-state Potts model partition functions and the equivalent Tutte polynomials for chain graphs comprised of m repeated hammock subgraphs \(H_{e_1,...,e_r}\) connected with line graphs of length \(e_g\) edges, such that the chains have open or cyclic boundary conditions (BC). Here, \(H_{e_1,...,e_r}\) is a hammock (series-parallel) subgraph with r separate paths along “ropes” with respective lengths \(e_1, ..., e_r\) edges, connecting the two end vertices. We denote the resultant chain graph as \(G_{\{e_1,...,e_r\},e_g,m;BC}\). We discuss special cases, including chromatic, flow, and reliability polynomials. In the case of cyclic boundary conditions, the zeros of the Potts partition function in the complex q function accumulate, in the limit \(m \rightarrow \infty \), onto curves forming a locus \(\mathcal{B}\), and we study this locus.