Chromatic Polynomials for $J(\prod H)I$ Strip Graphs and their Asymptotic Limits

Abstract

We calculate the chromatic polynomials $P$ for $n$-vertex strip graphs of the form $J({\prod }_{\ell =1}^{m}H)I$, where $J$ and $I$ are various subgraphs on the left and right ends of the strip, whose bulk is comprised of $m$-fold repetitions of a subgraph $H$. The strips have free boundary conditions in the longitudinal direction and free or periodic boundary conditions in the transverse direction. This extends our earlier calculations for strip graphs of the form $({\prod }_{\ell =1}^{m}H)I$. We use a generating function method. From these results we compute the asymptotic limiting function $W={\lim }_{n\to \infty }{P}^{1/n}$; for $q\in {\mathbb{Z}}_{+}$ this has physical significance as the ground state degeneracy per site (exponent of the ground state entropy) of the $q$-state Potts antiferromagnet on the given strip. In the complex $q$ plane, $W$ is an analytic function except on a certain continuous locus $\mathcal{B}$. In contrast to the $({\prod }_{\ell =1}^{m}H)I$ strip graphs, where $\mathcal{B}$ (i) is independent of $I$, and (ii) consists of arcs and possible line segments that do not enclose any regions in the $q$ plane, we find that for some $J({\prod }_{\ell =1}^{m}H)I$ strip graphs, $\mathcal{B}$ (i) does depend on $I$ and $J$, and (ii) can enclose regions in the $q$ plane. Our study elucidates the effects of different end subgraphs $I$ and $J$ and of boundary conditions on the infinite-length limit of the strip graphs.

33 pages, Latex, 7 encapsulated postscript figures, Physica A, in press, with some typos fixed

References in corpus (6)

• Chromatic Polynomials for Families of Strip Graphs and their Asymptotic Limits
• Lower Bounds and Series for the Ground State Entropy of the Potts Antiferromagnet on Archimedean Lattices and their Duals
• Ground State Entropy of Potts Antiferromagnets on Homeomorphic Families of Strip Graphs
• Families of Graphs with W_r({G},q) Functions That Are Nonanalytic at 1/q=0
• Families of Graphs With Chromatic Zeros Lying on Circles
• End Graph Effects on Chromatic Polynomials for Strip Graphs of Lattices and their Asymptotic Limits

Cited by in corpus (35)

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• Thin Fisher Zeroes
• Exact Potts Model Partition Functions on Wider Arbitrary-Length Strips of the Square Lattice
• Exact Potts Model Partition Function on Strips of the Triangular Lattice
• T=0 Partition Functions for Potts Antiferromagnets on Square Lattice Strips with (Twisted) Periodic Boundary Conditions
• Structural Properties of Potts Model Partition Functions and Chromatic Polynomials for Lattice Strips
• Ground State Entropy of Potts Antiferromagnets: Cases with Noncompact W Boundaries Having Multiple Points at 1/q = 0
• Ground-State Degeneracy of Potts Antiferromagnets on Two-Dimensional Lattices: Approach Using Infinite Cyclic Strip Graphs
• 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
• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models III. Triangular-Lattice Chromatic Polynomial
• Ground State Entropy of the Potts Antiferromagnet on Cyclic Strip Graphs
• Ground State Entropy of Potts Antiferromagnets on Cyclic Polygon Chain Graphs
• Complex-Temperature Phase Diagrams for the q-State Potts Model on Self-Dual Families of Graphs and the Nature of the $q\to \infty$ Limit
• $q$-Plane Zeros of the Potts Partition Function on Diamond Hierarchical Graphs
• Zeros of Jones Polynomials for Families of Knots and Links
• Potts Model Partition Functions for Self-Dual Families of Strip Graphs
• Ground State Entropy of the Potts Antiferromagnet with Next-Nearest-Neighbor Spin-Spin Couplings on Strips of the Square Lattice
• T=0 Partition Functions for Potts Antiferromagnets on Lattice Strips with Fully Periodic Boundary Conditions
• Ground State Entropy of the Potts Antiferromagnet on Triangular Lattice Strips
• 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
• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial
• Transfer matrices and partition-function zeros for antiferromagnetic Potts models. VI. Square lattice with special boundary conditions
• General Structural Results for Potts Model Partition Functions on Lattice Strips
• Potts model on recursive lattices: some new exact results
• Chromatic Polynomials for Lattice Strips with Cyclic Boundary Conditions
• Complex-q zeros of the partition function of the Potts model with long-range interactions
• Planar triangulations with real chromatic roots arbitrarily close to four
• Transfer Matrices for the Zero-Temperature Potts Antiferromagnet on Cyclic and Mobius Lattice Strips
• Exact Chromatic Polynomials for Toroidal Chains of Complete Graphs
• End Graph Effects on Chromatic Polynomials for Strip Graphs of Lattices and their Asymptotic Limits