Spanning trees on lattices and integral identities

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(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 identities can be obtained. We also gi...

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 NST 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 2 dimensions, and is applied to the hypercubic, body-centred cubic, face-ce...

Spanning trees on hypercubic lattices and nonorientable surfaces

We consider the problem of enumerating spanning trees on lattices. Closed-form expressions are obtained for the spanning tree generating function for a hypercubic lattice of size N1×N2×· · ·×Nd in d dimensions under free, periodic, and a combination of free and periodic boundary conditions. Results are also obtained for a simple quartic net embedded on two non-orientable surfaces, a Möbius stri...

Spanning trees in subgraphs of lattices

Spanning Trees of Lattices Embedded on the Klein Bottle

The problem of enumerating spanning trees in lattices with Klein bottle boundary condition is considered here. The exact closed-form expressions of the numbers of spanning trees for 4.8.8 lattice, hexagonal lattice, and 3(3) · 4(2) lattice on the Klein bottle are presented.