Coverings, heat kernels and spanning trees

Abstract

We consider a graph G and a covering G̃ of G and we study the relations of their eigenvalues and heat kernels. We evaluate the heat kernel for an infinite k-regular tree and we examine the heat kernels for general k-regular graphs. In particular, we show that a k-regular graph on n vertices has at most (Equation Presented) spanning trees, which is best possible within a constant factor.