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 greater than or equal to 2 dimensions, and is applied to the hypercubic, body-centred cubic, face-centred cubic and specific planar lattices including the kagome, 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 L with the property that N-ST similar to exp(nz(L)) as the number of vertices n --> infinity, where z(L) is a finite non-zero 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(L) exactly for the lattices we consider, and discuss the dependence of z(L) on d and the lattice coordination number. Pie also establish a relation connecting z(L) to the free energy of the critical Ising model for planar lattices.