Shu Chiuan Chang

Spanning trees on the Sierpinski gasket

We present the numbers of spanning trees on the Sierpinski gasket SGd(n) at stage n with dimension d equal to two, three and four. The general expression for the number of spanning trees on SGd(n) with arbitrary d is conjectured. The numbers of spanning trees on the generalized Sierpinski gasket SGd,b(n) with d = 2 and b = 3,4 are also obtained.

Exact Potts model partition function on strips of the triangular lattice

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 Ly=2 and arbitrary length, of the triangular lattice with free, cyclic, and Mobius longitudinal boundary conditions. These partition functions are equivalent to Tutte/Whitney polynomials for these graphs. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. Considering the full generalization to arbitrary complex q and temperature, we determine the singular locus B in the corresponding C2 space, arising as the accumulation set of partition function zeros as n→∞. In particular, we study the connection with the T=0 limit of the Potts antiferromagnet where B reduces to the accumulation set of chromatic zeros. Comparisons are made with our previous exact calculation of Potts model partition functions for the corresponding strips of the square lattice. Our present calculations yield, as special cases, several quantities of graph-theoretic interest.

Exact Potts model partition functions on wider arbitrary-length strips of the square lattice

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 Ly=3 vertices and arbitrary length Lx with periodic longitudinal boundary conditions, of the following types: (i) (FBCy,PBCx)= cyclic, (ii) (FBCy,TPBCx)= Mobius, (iii) (PBCy,PBCx)= toroidal, and (iv) (PBCy,TPBCx)= Klein bottle, where FBC and (T)PBC refer to free and (twisted) periodic boundary conditions. Results for the Ly=2 torus and Klein bottle strips are also included. In the infinite-length limit the thermodynamic properties are discussed and some general results are given for low-temperature behavior on strips of arbitrarily great width. We determine the submanifold in the C2 space of q and temperature where the free energy is singular for these strips. Our calculations are also used to compute certain quantities of graph-theoretic interest.

Zeros of Jones polynomials for families of knots and links

We calculate Jones polynomials VL(t) for several families of alternating knots and links by computing the Tutte polynomials T(G,x,y) for the associated graphs G and then obtaining VL(t) as a special case of the Tutte polynomial. For each of these families we determine the zeros of the Jones polynomial, including the accumulation set in the limit of infinitely many crossings. A discussion is also given of the calculation of Jones polynomials for non-alternating links.

Structural properties of Potts model partition functions and chromatic polynomials for lattice strips

The q-state Potts model partition function (equivalent to the Tutte polynomial) for a lattice strip of fixed width Ly and arbitrary length Lx has the form Z(G,q,v)=∑j=1NZ,G,λcZ,G,j(λZ,G,j)Lx, where v is a temperature-dependent variable. The special case of the zero-temperature antiferromagnet (v=−1) is the chromatic polynomial P(G,q). Using coloring and transfer matrix methods, we give general formulas for CX,G=∑j=1NX,G,λcX,G,j for X=Z,P on cyclic and Mobius strip graphs of the square and triangular lattice. Combining these with a general expression for the (unique) coefficient cZ,G,j of degree d in q: c(d)=U2d(q/2), where Un(x) is the Chebyshev polynomial of the second kind, we determine the number of λZ,G,js with coefficient c(d) in Z(G,q,v) for these cyclic strips of width Ly to be nZ(Ly,d)=(2d+1)(Ly+d+1)−12LyLy−d for 0⩽d⩽Ly and zero otherwise. For both cyclic and Mobius strips of these lattices, the total number of distinct eigenvalues λZ,G,j is calculated to be NZ,Ly,λ=2LyLy. Results are also presented for the analogous numbers nP(Ly,d) and NP,Ly,λ for P(G,q). We find that nP(Ly,0)=nP(Ly−1,1)=MLy−1 (Motzkin number), nZ(Ly,0)=CLy (the Catalan number), and give an exact expression for NP,Ly,λ. Our results for NZ,Ly,λ and NP,Ly,λ apply for both the cyclic and Mobius strips of both the square and triangular lattices; we also point out the interesting relations NZ,Ly,λ=2NDA,tri,Ly and NP,Ly,λ=2NDA,sq,Ly, where NDA,Λ,n denotes the number of directed lattice animals on the lattice Λ. We find the asymptotic growths NZ,Ly,λ∼Ly−1/24Ly and NP,Ly,λ∼Ly−1/23Ly as Ly→∞. Some general geometric identities for Potts model partition functions are also presented.

Exact Potts Model Partition Functions for Strips of the Triangular Lattice

AbstractWe 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 transverse boundary conditions. These partition functions have the form Z(G,q,v)=

Ground state entropy of the Potts antiferromagnet on strips of the square lattice

\sum _{j = 1}^{N_{Z,G,\lambda } }

Some exact results for spanning trees on lattices

cz,G,j(λz,G,j)m-1. We give general formulas for NZ,G,j and its specialization to v=−1 for arbitrary L. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite triangular lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus