arXiv subjects

Shu-Chiuan Chang

Publications and source records attributed to Shu-Chiuan Chang.

At least 37 records · Page 2Linked to original sources

arXiv2009

Weighted Graph Colorings

We study two weighted graph coloring problems, in which one assigns $q$ colors to the vertices of a graph such that adjacent vertices have different colors, with a vertex weighting $w$ that either disfavors or favors a given color. We exhibit a weighted chromatic polynomial $Ph(G,q,w)$ associated with this problem that generalizes the chromatic polynomial $P(G,q)$. General properties of this polynomial are proved, and illustrative calculations for various families of graphs are presented. We show that the weighted chromatic polynomial is able to distinguish between certain graphs that yield the same chromatic polynomial. We give a general structural formula for $Ph(G,q,w)$ for lattice strip graphs $G$ with periodic longitudinal boundary conditions. The zeros of $Ph(G,q,w)$ in the $q$ and $w$ planes and their accumulation sets in the limit of infinitely many vertices of $G$ are analyzed. Finally, some related weighted graph coloring problems are mentioned.

math-ph↗

arXiv2009

Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips

We determine the general structure of the partition function of the $q$-state Potts model in an external magnetic field, $Z(G,q,v,w)$ for arbitrary $q$, temperature variable $v$, and magnetic field variable $w$, on cyclic, Möbius, and free strip graphs $G$ of the square (sq), triangular (tri), and honeycomb (hc) lattices with width $L_y$ and arbitrarily great length $L_x$. For the cyclic case we prove that the partition function has the form $Z(Λ,L_y \times L_x,q,v,w)=\sum_{d=0}^{L_y} \tilde c^{(d)} Tr[(T_{Z,Λ,L_y,d})^m]$, where $Λ$ denotes the lattice type, $\tilde c^{(d)}$ are specified polynomials of degree $d$ in $q$, $T_{Z,Λ,L_y,d}$ is the corresponding transfer matrix, and $m=L_x$ ($L_x/2$) for $Λ=sq, tri (hc)$, respectively. An analogous formula is given for Möbius strips, while only $T_{Z,Λ,L_y,d=0}$ appears for free strips. We exhibit a method for calculating $T_{Z,Λ,L_y,d}$ for arbitrary $L_y$ and give illustrative examples. Explicit results for arbitrary $L_y$ are presented for $T_{Z,Λ,L_y,d}$ with $d=L_y$ and $d=L_y-1$. We find very simple formulas for the determinant $det(T_{Z,Λ,L_y,d})$. We also give results for self-dual cyclic strips of the square lattice.

cond-mat.stat-mech↗

arXiv2009

Some Exact Results on the Potts Model Partition Function in a Magnetic Field

We consider the Potts model in a magnetic field on an arbitrary graph $G$. Using a formula of F. Y. Wu for the partition function $Z$ of this model as a sum over spanning subgraphs of $G$, we prove some properties of $Z$ concerning factorization, monotonicity, and zeros. A generalization of the Tutte polynomial is presented that corresponds to this partition function. In this context we formulate and discuss two weighted graph-coloring problems. We also give a general structural result for $Z$ for cyclic strip graphs.

cond-mat.stat-mech↗

arXiv2008

Spanning Trees on the Two-Dimensional Lattices with More Than One Type of Vertex

For a two-dimensional lattice $Λ$ with $n$ vertices, the number of spanning trees $N_{ST}(Λ)$ grows asymptotically as $\exp(n z_Λ)$ in the thermodynamic limit. We present exact integral expression and numerical value for the asymptotic growth constant $z_Λ$ for spanning trees on various two-dimensional lattices with more than one type of vertex given in \cite{Okeeffe}. An exact closed-form expression for the asymptotic growth constant is derived for net 14, and the asymptotic growth constants of net 27 and the triangle lattice have the simple relation $z_{27} = (z_{tri}+\ln 4)/4$. Some integral identities are also obtained.

cond-mat.stat-mech↗

arXiv2008

Number of connected spanning subgraphs on the Sierpinski gasket

We study the number of connected spanning subgraphs $f_{d,b}(n)$ on the generalized Sierpinski gasket $SG_{d,b}(n)$ at stage $n$ with dimension $d$ equal to two, three and four for $b=2$, and layer $b$ equal to three and four for $d=2$. The upper and lower bounds for the asymptotic growth constant, defined as $z_{SG_{d,b}}=\lim_{v \to \infty} \ln f_{d,b}(n)/v$ where $v$ is the number of vertices, on $SG_{2,b}(n)$ with $b=2,3,4$ are derived in terms of the results at a certain stage. The numerical values of $z_{SG_{d,b}}$ are obtained.

math-ph↗

arXiv2008

Structure of spanning trees on the two-dimensional Sierpinski gasket

Consider spanning trees on the two-dimensional Sierpinski gasket SG(n) where stage $n$ is a non-negative integer. For any given vertex $x$ of SG(n), we derive rigorously the probability distribution of the degree $j \in \{1,2,3,4\}$ at the vertex and its value in the infinite $n$ limit. Adding up such probabilities of all the vertices divided by the number of vertices, we obtain the average probability distribution of the degree $j$. The corresponding limiting distribution $ϕ_j$ gives the average probability that a vertex is connected by 1, 2, 3 or 4 bond(s) among all the spanning tree configurations. They are rational numbers given as $ϕ_1=10957/40464$, $ϕ_2=6626035/13636368$, $ϕ_3=2943139/13636368$, $ϕ_4=124895/4545456$.

math-ph↗

arXiv2007

Dimer coverings on the Sierpinski gasket with possible vacancies on the outmost vertices

We present the number of dimers $N_d(n)$ on the Sierpinski gasket $SG_d(n)$ at stage $n$ with dimension $d$ equal to two, three, four or five, where one of the outmost vertices is not covered when the number of vertices $v(n)$ is an odd number. The entropy of absorption of diatomic molecules per site, defined as $S_{SG_d}=\lim_{n \to \infty} \ln N_d(n)/v(n)$, is calculated to be $\ln(2)/3$ exactly for $SG_2(n)$. The numbers of dimers on the generalized Sierpinski gasket $SG_{d,b}(n)$ with $d=2$ and $b=3,4,5$ are also obtained exactly. Their entropies are equal to $\ln(6)/7$, $\ln(28)/12$, $\ln(200)/18$, respectively. The upper and lower bounds for the entropy are derived in terms of the results at a certain stage for $SG_d(n)$ with $d=3,4,5$. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of $S_{SG_d}$ with $d=3,4,5$ can be evaluated with more than a hundred significant figures accurate.

cond-mat.stat-mech↗

arXiv2007

Exact Potts Model Partition Functions for Strips of the Honeycomb Lattice

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 honeycomb lattice for a variety of transverse widths equal to $L_y$ vertices and for arbitrarily great length, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These partition functions have the form $Z(G,q,v)=\sum_{j=1}^{N_{Z,G,λ}} c_{Z,G,j}(λ_{Z,G,j})^m$, where $m$ denotes the number of repeated subgraphs in the longitudinal direction. We give general formulas for $N_{Z,G,j}$ for arbitrary $L_y$. We also present plots of zeros of the partition function in the $q$ plane for various values of $v$ and in the $v$ plane for various values of $q$. Explicit results for partition functions are given in the text for $L_y=2,3$ (free) and $L_y=4$ (cylindrical), and plots of partition function zeros are given for $L_y$ up to 5 (free) and $L_y=6$ (cylindrical). Plots of the internal energy and specific heat per site for infinite-length strips are also presented.

cond-mat.stat-mech↗

arXiv2007

Dimer-monomer model on the Sierpinski gasket

We present the numbers of dimer-monomers on the Sierpinski gasket $SG_d(n)$ at stage $n$ with dimension $d$ equal to two, three and four, and determine the asymptotic behaviors. The corresponding results on the generalized Sierpinski gasket $SG_{d,b}(n)$ with $d=2$ and $b=3,4$ are obtained.

cond-mat.stat-mech↗

arXiv2006

Spanning forests on the Sierpinski gasket

We present the numbers of spanning forests on the Sierpinski gasket $SG_d(n)$ at stage $n$ with dimension $d$ equal to two, three and four, and determine the asymptotic behaviors. The corresponding results on the generalized Sierpinski gasket $SG_{d,b}(n)$ with $d=2$ and $b=3,4$ are obtained. We also derive the upper bounds of the asymptotic growth constants for both $SG_d$ and $SG_{2,b}$.

math-ph↗

arXiv2006

Spanning trees on the Sierpinski gasket

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

cond-mat.stat-mech↗

arXiv2006

Spanning Trees on Lattices and Integration Identities

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 $\exp(n z_Λ)$ 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 identities can be obtained. We also give $z_{Λ(p)}$ on the homeomorphic expansion of $k$-regular lattices with $p$ vertices inserted on each edge.

cond-mat.stat-mech↗

arXiv2006

Some Exact Results for Spanning Trees on Lattices

For $n$-vertex, $d$-dimensional lattices $Λ$ with $d \ge 2$, the number of spanning trees $N_{ST}(Λ)$ grows asymptotically as $\exp(n z_Λ)$ in the thermodynamic limit. We present an exact closed-form result for the asymptotic growth constant $z_{bcc(d)}$ for spanning trees on the $d$-dimensional body-centered cubic lattice. We also give an exact integral expression for $z_{fcc}$ on the face-centered cubic lattice and an exact closed-form expression for $z_{488}$ on the $4 \cdot 8 \cdot 8$ lattice.

cond-mat.stat-mech↗

arXiv2006

Partition Function Zeros of a Restricted Potts Model on Lattice Strips and Effects of Boundary Conditions

We calculate the partition function $Z(G,Q,v)$ of the $Q$-state Potts model exactly for strips of the square and triangular lattices of various widths $L_y$ and arbitrarily great lengths $L_x$, with a variety of boundary conditions, and with $Q$ and $v$ restricted to satisfy conditions corresponding to the ferromagnetic phase transition on the associated two-dimensional lattices. From these calculations, in the limit $L_x \to \infty$, we determine the continuous accumulation loci ${\cal B}$ of the partition function zeros in the $v$ and $Q$ planes. Strips of the honeycomb lattice are also considered. We discuss some general features of these loci.

cond-mat.stat-mech↗

arXiv2006

Partition Function Zeros of a Restricted Potts Model on Self-Dual Strips of the Square Lattice

We calculate the partition function $Z(G,Q,v)$ of the $Q$-state Potts model exactly for self-dual cyclic square-lattice strips of various widths $L_y$ and arbitrarily great lengths $L_x$, with $Q$ and $v$ restricted to satisfy the relation $Q=v^2$. From these calculations, in the limit $L_x \to \infty$, we determine the continuous accumulation locus ${\cal B}$ of the partition function zeros in the $v$ and $Q$ planes. A number of interesting features of this locus are discussed and a conjecture is given for properties applicable for arbitrarily great width. Relations with the loci ${\cal B}$ for general $Q$ and $v$ are analyzed.

cond-mat.stat-mech↗

arXiv2005

Zeros of the Potts Model Partition Function in the Large-$q$ Limit

We study the zeros of the $q$-state Potts model partition function $Z(Λ,q,v)$ for large $q$, where $v$ is the temperature variable and $Λ$ is a section of a regular $d$-dimensional lattice with coordination number $κ_Λ$ and various boundary conditions. We consider the simultaneous thermodynamic limit and $q \to \infty$ limit and show that when these limits are taken appropriately, the zeros lie on the unit circle $|x_Λ|=1$ in the complex $x_Λ$ plane, where $x_Λ=v q^{-2/κ_Λ}$. For large finite sections of some lattices we also determine the circular loci near which the zeros lie for large $q$.

cond-mat.stat-mech↗

arXiv2005

Transfer Matrices for the Partition Function of the Potts Model on Toroidal Lattice Strips

We present a method for calculating transfer matrices for the $q$-state Potts model partition functions $Z(G,q,v)$, for arbitrary $q$ and temperature variable $v$, on strip graphs $G$ of the square (sq), triangular (tri), and honeycomb (hc) lattices of width $L_y$ vertices and of arbitrarily great length $L_x$ vertices, subject to toroidal and Klein bottle boundary conditions. For the toroidal case we express the partition function as $Z(Λ, L_y \times L_x,q,v) = \sum_{d=0}^{L_y} \sum_j b_j^{(d)} (λ_{Z,Λ,L_y,d,j})^m$, where $Λ$ denotes lattice type, $b_j^{(d)}$ are specified polynomials of degree $d$ in $q$, $λ_{Z,Λ,L_y,d,j}$ are eigenvalues of the transfer matrix $T_{Z,Λ,L_y,d}$ in the degree-$d$ subspace, and $m=L_x$ ($L_x/2$) for $Λ=sq, tri (hc)$, respectively. An analogous formula is given for Klein bottle strips. We exhibit a method for calculating $T_{Z,Λ,L_y,d}$ for arbitrary $L_y$. In particular, we find some very simple formulas for the determinant $det(T_{Z,Λ,L_y,d})$, and trace $Tr(T_{Z,Λ,L_y})$. Corresponding results are given for the equivalent Tutte polynomials for these lattice strips and illustrative examples are included.

cond-mat.stat-mech↗

arXiv2004

Transfer Matrices for the Zero-Temperature Potts Antiferromagnet on Cyclic and Mobius Lattice Strips

We present transfer matrices for the zero-temperature partition function of the $q$-state Potts antiferromagnet (equivalently, the chromatic polynomial) on cyclic and Möbius strips of the square, triangular, and honeycomb lattices of width $L_y$ and arbitrarily great length $L_x$. We relate these results to our earlier exact solutions for square-lattice strips with $L_y=3,4,5$, triangular-lattice strips with $L_y=2,3,4$, and honeycomb-lattice strips with $L_y=2,3$ and periodic or twisted periodic boundary conditions. We give a general expression for the chromatic polynomial of a Möbius strip of a lattice $Λ$ and exact results for a subset of honeycomb-lattice transfer matrices, both of which are valid for arbitrary strip width $L_y$. New results are presented for the $L_y=5$ strip of the triangular lattice and the $L_y=4$ and $L_y=5$ strips of the honeycomb lattice. Using these results and taking the infinite-length limit $L_x \to \infty$, we determine the continuous accumulation locus of the zeros of the above partition function in the complex $q$ plane, including the maximal real point of nonanalyticity of the degeneracy per site, $W$ as a function of $q$.

cond-mat.stat-mech↗