Spanning Trees and Mahler Measure

Abstract

The complexity of a finite connected graph is its number of spanning trees; for a non-connected graph it is the product of complexities of its connected components. If $G$ is an infinite graph with cofinite free ${\mathbb Z}^d$-symmetry, then the logarithmic Mahler measure $m(\Delta)$ of its Laplacian polynomial $\Delta$ is the exponential growth rate of the complexity of finite quotients of $G$. It is bounded below by $m(\Delta({\mathbb G}_d))$, where ${\mathbb G}_d$ is the grid graph of dimension $d$. The growth rates $m(\Delta({\mathbb G}_d))$ are asymptotic to $\log 2d$ as $d$ tends to infinity. If $m(\Delta(G))\ne 0$, then $m(\Delta(G)) \ge \log 2$. An application to determinant growth rates of families of alternating links arising from planar graphs is given.