Coverings, Heat Kernels and Spanning Trees

We consider a graph $G$ and a covering $\tilde{G}$ 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 $$ (1+o(1)) {{2\log n}\over {kn \log k}} \left( {{ (k-1)^{k-1}}\over {(k^2-2k)^{k/2-1}}}\right)^n $$ spanning trees, which is best possible within a constant factor.