Chromatic Polynomials for Lattice Strips with Cyclic Boundary Conditions

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 $P(G,q)={\sum }_{j=1}^{N_{G,\lambda }}{c}_{G,j}({\lambda }_{G,j}{)}^{L_x}$, and is equivalent to the chromatic polynomial for this graph. We present exact zero-temperature partition functions for strips of several lattices with $(FB{C}_y,PB{C}_x)$, 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 $c^{(d)}=U_{2d}(\frac{\sqrt{q}}{2})$, 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 $(PB{C}_y,PB{C}_x)$, 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, $\mathcal{B}$ are found.

41 pages, latex, 18 figures

References in corpus (8)

• Exact Potts Model Partition Functions on Ladder Graphs
• Exact Potts Model Partition Function on Strips of the Triangular Lattice
• Structural Properties of Potts Model Partition Functions and Chromatic Polynomials for Lattice Strips
• T=0 Partition Functions for Potts Antiferromagnets on Moebius Strips and Effects of Graph Topology
• Ground State Entropy of the Potts Antiferromagnet on Strips of the Square Lattice
• Exact Potts Model Partition Functions on Strips of the Honeycomb Lattice
• Ground State Entropy of Potts Antiferromagnets on Cyclic Polygon Chain Graphs
• T=0 Partition Functions for Potts Antiferromagnets on Lattice Strips with Fully Periodic Boundary Conditions

Cited by in corpus (2)

• Transfer Matrices for the Zero-Temperature Potts Antiferromagnet on Cyclic and Mobius Lattice Strips
• Exact Chromatic Polynomials for Toroidal Chains of Complete Graphs