### Abstract

The zero-temperature q-state Potts model partition function for a lattice strip of fixed width L_{y} and arbitrary length L_{x} has the form P(G,q) = ∑_{j = 1}^{NG, λ} c_{G, j}(λ_{G, j})^{Lx}, and is equivalent to the chromatic polynomial for this graph. We present exact zero-temperature partition functions for strips of several lattices with (FBC_{y}, PBC_{x}), i.e., cyclic, boundary conditions. In particular, the chromatic polynomial of a family of generalized dodecahedra graphs is calculated. The coefficient c_{G, j} of degree d in q is c^{(d)} = U_{2d}(√q/2), where U_{n}(x) is the Chebyshev polynomial of the second kind. We also present the chromatic polynomial for the strip of the square lattice with (PBC_{y}, PBC_{x}), i.e., toroidal, boundary conditions and width L_{y} = 4 with the property that each set of four vertical vertices forms a tetrahedron. A number of interesting and novel features of the continuous accumulation set of the chromatic zeros, B are found.

Original language | English |
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Pages (from-to) | 495-522 |

Number of pages | 28 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 296 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 2001 Jul 15 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Condensed Matter Physics