Exact Potts/Tutte Polynomials for Hammock Chain Graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909641317810176 |
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| author | Chen, Yue Shrock, Robert |
| author_facet | Chen, Yue Shrock, Robert |
| contents | 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 $m \to \infty$, onto curves forming a locus ${\cal B}$, and we study this locus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22430 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exact Potts/Tutte Polynomials for Hammock Chain Graphs Chen, Yue Shrock, Robert Statistical Mechanics 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 $m \to \infty$, onto curves forming a locus ${\cal B}$, and we study this locus. |
| title | Exact Potts/Tutte Polynomials for Hammock Chain Graphs |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2410.22430 |