Exact Potts/Tutte Polynomials for Hammock Chain Graphs

Abstract

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 $\mathcal{B}$, and we study this locus.

57 pages, latex, 26 figures

References in corpus (14)

• Chromatic roots are dense in the whole complex plane
• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models I. General Theory and Square-Lattice Chromatic Polynomial
• A Little Statistical Mechanics for the Graph Theorist
• Exact Potts Model Partition Functions on Wider Arbitrary-Length Strips of the Square Lattice
• Ground State Entropy of the Potts Antiferromagnet on Strips of the Square Lattice
• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models II. Extended Results for Square-Lattice Chromatic Polynomial
• Exact Potts Model Partition Functions on Strips of the Honeycomb Lattice
• Exact Potts Model Partition Functions for Strips of the Triangular Lattice
• Reliability Polynomials and their Asymptotic Limits for Families of Graphs
• Exact T=0 Partition Functions for Potts Antiferromagnets on Sections of the Simple Cubic Lattice
• On the chromatic roots of generalized theta graphs
• General Structural Results for Potts Model Partition Functions on Lattice Strips
• Exact Potts/Tutte Polynomials for Polygon Chain Graphs
• Asymptotic Behavior of Acyclic and Cyclic Orientations of Directed Lattice Graphs