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

TL;DR

This paper derives exact and numerical expressions for the asymptotic growth constant of spanning trees on various two-dimensional lattices with multiple vertex types, including new closed-form formulas and integral identities.

Contribution

It provides the first exact integral and closed-form expressions for the growth constants on complex lattices with multiple vertex types, expanding understanding of spanning trees in such structures.

Findings

Exact integral expression for growth constant on various lattices

Closed-form formula for net 14's growth constant

Relation between net 27 and triangle lattice growth constants

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.