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We present exact calculations of reliability polynomials R(G, p) for lattice strips G of fixed widths L y ≤ 4 and arbitrarily great length L x with various boundary conditions. We introduce the notion of a reliability per vertex, r({G}, p) = lim |V |→∞ R(G, p) 1/|V | where |V | denotes the number of vertices in G and {G} denotes the formal limit lim |V |→∞… (More)

For a lattice Λ with n vertices and dimension d equal or higher than two, the number of spanning trees N ST (Λ) grows asymptotically as exp(nz Λ) in the thermodynamic limit. We present exact integral expressions for the asymptotic growth constant z Λ for spanning trees on several lattices. By taking different unit cells in the calculation, many integration… (More)

We present exact calculations of the partition function of the q-state Potts model for general q and temperature on strips of the square lattice of width L y = 3 vertices and arbitrary length L x with periodic longitudinal boundary conditions, of the following types: (i) (F BC y , P BC x) = cyclic, (ii) (F BC y , T P BC x) = Möbius, (iii) (P BC y , P BC x)… (More)

We present exact solutions for the zero-temperature partition function (chromatic polynomial P) and the ground state degeneracy per site W (= exponent of the ground-state entropy) for the q-state Potts antiferromagnet on strips of the square lattice of width L y vertices and arbitrarily great length L x vertices. The specific solutions are for (a) L y = 4,… (More)

We present exact calculations of the Potts model partition function Z(G, q, v) for arbitrary q and temperature-like variable v on n-vertex strip graphs G of the triangular lattice for a variety of transverse widths equal to L vertices and for arbitrarily great length equal to m vertices, with free longitudinal boundary conditions and free and periodic… (More)

In this paper we present exact calculations of the partition function Z of the q-state Potts model and its generalization to real q, for arbitrary temperature on n-vertex strip graphs, of width L y = 2 and arbitrary length, of the triangular lattice with free, cyclic, and Möbius longitudinal boundary conditions. These partition functions are equivalent to… (More)

We present exact calculations of the partition function of the q-state Potts model on (i) open, (ii) cyclic, and (iii) Möbius strips of the honeycomb (brick) lattice of width L y = 2 and arbitrarily great length. In the infinite-length limit the thermodynamic properties are discussed. The continuous locus of singularities of the free energy is determined in… (More)

We present exact calculations of the Potts model partition function Z(G, q, v) for arbitrary q and temperature-like variable v on n-vertex square-lattice strip graphs G for a variety of transverse widths L t and for arbitrarily great length L ℓ , with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These have… (More)

We present exact calculations of the zero-temperature partition function for the q-state Potts antiferromagnet (equivalently, the chromatic polynomial) for families of arbitrarily long strip graphs of the square and triangular lattices with width L y = 4 and boundary conditions that are doubly periodic or doubly periodic with reversed orientation (i.e. of… (More)