Spanning Trees on Hypercubic Lattices and Non-orientable Surfaces

Details Spanning Trees on Hypercubic Lattices and Non-orientable Surfaces Spanning Trees on Hypercubic Lattices and Non-orientable Surfaces

2000-01-27

arxiv.org/abs/cond-mat/0001408v1

Appl. Math. Letters 13 (6), 19-25 (2000).

Physics

Condensed Matter

Statistical Mechanics

latex, 9 pages, no figures, to appear in Lett. Appl. Math

Scientific paper

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 N_1 x N_2 x...x N_d 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 Moebius strip and the Klein bottle. Our results are based on the use of a formula expressing the spanning tree generating function in terms of the eigenvalues of an associated tree matrix. An elementary derivation of this formula is given.

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