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 $H_{e_1,\mathrm{.}\mathrm{.}\mathrm{.},e_r}$ connected with line graphs of length $e_g$ edges, such that the chains have open or cyclic boundary conditions (BC). Here, $H_{e_1,\mathrm{.}\mathrm{.}\mathrm{.},e_r}$ is a hammock (series-parallel) subgraph with $r$ separate paths along ``ropes’’ with respective lengths $e_1,\mathrm{.}\mathrm{.}\mathrm{.},e_r$ edges, connecting the two end vertices. We denote the resultant chain graph as $G_{\{e_1,\mathrm{.}\mathrm{.}\mathrm{.},e_r\},e_g,m;BC}$. 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 $m\to \mathrm{\infty }$, onto curves forming a locus $\mathcal{B}$, and we study this locus.