Exact Potts/Tutte Polynomials for Hammock Chain Graphs

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 He1,..,er connected with line graphs of length eg edges, such that the chains have open or cyclic boundary conditions (BC). Here, He1,..,er is a hammock (series-parallel) subgraph with r separate paths along “ropes” with respective lengths e1,..,er edges, connecting the two end vertices. We denote the resultant chain graph as G{e1,..,er},eg,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→∞, onto curves forming a locus B, and we study this locus.