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-centred cubic and specific planar lattices including the kagomé, diced, 4–8–8 (bathroom-tile), Union Jack and 3–12–12 lattices. This leads to closed-form expressions for NST for these lattices of finite sizes. We prove a theorem concerning the classes of graphs and lattices L with the property that NST ∼ exp(nzL) as the number of vertices n → ∞, where zL 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 zL exactly for the lattices we consider, and discuss the dependence of zL on d and the lattice coordination number. We also establish a relation connecting zL to the free energy of the critical Ising model for planar lattices.