The Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary

We consider the dimer-monomer problem for the rectangular lattice. By mapping the problem into one of close-packed dimers on an extended lattice, we rederive the Tzeng-Wu solution for a single monomer on the boundary by evaluating a Pfaffian. We also clarify the mathematical content of the Tzeng-Wu solution by identifying it as the product of the nonzero eigenvalues of the Kasteleyn matrix.Comment: 4 Pages to appear in the Physical Review E (2006

Theory of impedance networks: The two-point impedance and LC resonances

We present a formulation of the determination of the impedance between any two nodes in an impedance network. An impedance network is described by its Laplacian matrix L which has generally complex matrix elements. We show that by solving the equation L u_a = lambda_a u_a^* with orthonormal vectors u_a, the effective impedance between nodes p and q of the network is Z = Sum_a [u_{a,p} - u_{a,q}]^2/lambda_a where the summation is over all lambda_a not identically equal to zero and u_{a,p} is the p-th component of u_a. For networks consisting of inductances (L) and capacitances (C), the formulation leads to the occurrence of resonances at frequencies associated with the vanishing of lambda_a. This curious result suggests the possibility of practical applications to resonant circuits. Our formulation is illustrated by explicit examples.Comment: 21 pages, 3 figures; v4: typesetting corrected; v5: Eq. (63) correcte

Theory of resistor networks: The two-point resistance

The resistance between arbitrary two nodes in a resistor network is obtained in terms of the eigenvalues and eigenfunctions of the Laplacian matrix associated with the network. Explicit formulas for two-point resistances are deduced for regular lattices in one, two, and three dimensions under various boundary conditions including that of a Moebius strip and a Klein bottle. The emphasis is on lattices of finite sizes. We also deduce summation and product identities which can be used to analyze large-size expansions of two-and-higher dimensional lattices.Comment: 30 pages, 5 figures now included; typos in Example 1 correcte

Spanning Trees on Graphs and Lattices in d Dimensions

The problem of enumerating spanning trees on graphs and lattices is considered. We obtain bounds on the number of spanning trees $N_{ST}$ and establish inequalities relating the numbers of spanning trees of different graphs or lattices. A general formulation is presented for the enumeration of spanning trees on lattices in $d\geq 2$ dimensions, and is applied to the hypercubic, body-centered cubic, face-centered cubic, and specific planar lattices including the kagom\'e, diced, 4-8-8 (bathroom-tile), Union Jack, and 3-12-12 lattices. This leads to closed-form expressions for $N_{ST}$ for these lattices of finite sizes. We prove a theorem concerning the classes of graphs and lattices ${\cal L}$ with the property that $N_{ST} \sim \exp(nz_{\cal L})$ as the number of vertices $n \to \infty$, where $z_{\cal L}$ is a finite nonzero constant. This includes the bulk limit of lattices in any spatial dimension, and also sections of lattices whose lengths in some dimensions go to infinity while others are finite. We evaluate $z_{\cal L}$ exactly for the lattices we considered, and discuss the dependence of $z_{\cal L}$ on d and the lattice coordination number. We also establish a relation connecting $z_{\cal L}$ to the free energy of the critical Ising model for planar lattices ${\cal L}$.Comment: 28 pages, latex, 1 postscript figure, J. Phys. A, in pres

The two-point resistance of a resistor network: A new formulation and application to the cobweb network

We consider the problem of two-point resistance in a resistor network previously studied by one of us [F. Y. Wu, J. Phys. A {\bf 37}, 6653 (2004)]. By formulating the problem differently, we obtain a new expression for the two-point resistance between two arbitrary nodes which is simpler and can be easier to use in practice. We apply the new formulation to the cobweb resistor network to obtain the resistance between two nodes in the network. Particularly, our results prove a recently proposed conjecture on the resistance between the center node and a node on the network boundary. Our analysis also solves the spanning tree problem on the cobweb network.Comment: 14 pages, 1 figure, adding one references and discussions of special case

Ising model on nonorientable surfaces: Exact solution for the Moebius strip and the Klein bottle

Closed-form expressions are obtained for the partition function of the Ising model on an M x N simple-quartic lattice embedded on a Moebius strip and a Klein bottle for finite M and N. The finite-size effects at criticality are analyzed and compared with those under cylindrical and toroidal boundary conditions. Our analysis confirms that the central charge is c=1/2.Comment: 8 pages, 3 eps figure

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.Comment: 20 pages, 8 figures, 1 tabl

Uniform tiling with electrical resistors

The electric resistance between two arbitrary nodes on any infinite lattice structure of resistors that is a periodic tiling of space is obtained. Our general approach is based on the lattice Green's function of the Laplacian matrix associated with the network. We present several non-trivial examples to show how efficient our method is. Deriving explicit resistance formulas it is shown that the Kagom\'e, the diced and the decorated lattice can be mapped to the triangular and square lattice of resistors. Our work can be extended to the random walk problem or to electron dynamics in condensed matter physics.Comment: 22 pages, 14 figure