Coverings, Heat Kernels and Spanning Trees

@article{Graham1999CoveringsHK,
  title={Coverings, Heat Kernels and Spanning Trees},
  author={Fan Chung Graham and Shing-Tung Yau},
  journal={Electr. J. Comb.},
  year={1999},
  volume={6}
}
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 (1 + o(1)) 2 log n kn log k ! (k − 1)k−1 (k2 − 2k)k/2−1 "n spanning trees, which is best possible within a constant factor. 
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