Spanning Trees on the Two-Dimensional Lattices with More Than One Type of Vertex

Abstract

For a two-dimensional lattice $\Lambda$ with $n$ vertices, the number of spanning trees $N_{ST}(\Lambda)$ grows asymptotically as $\exp(n z_\Lambda)$ in the thermodynamic limit. We present exact integral expression and numerical value for the asymptotic growth constant $z_\Lambda$ for spanning trees on various two-dimensional lattices with more than one type of vertex given in \cite{Okeeffe}. An exact closed-form expression for the asymptotic growth constant is derived for net 14, and the asymptotic growth constants of net 27 and the triangle lattice have the simple relation $z_{27} = (z_{tri}+\ln 4)/4$. Some integral identities are also obtained.