Geodesic
Coverings, heat kernels and spanning trees
The electronic journal of combinatorics, Tome 6 (1999)
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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.
DOI : 10.37236/1444
Classification : 05C50, 35P05, 58J99
Mots-clés : covering, Laplacian, eigenvalues, heat kernel, number of spanning trees, bounds 
@article{10_37236_1444,
     author = {Fan Chung and S.-T. Yau},
     title = {Coverings, heat kernels and spanning trees},
     journal = {The electronic journal of combinatorics},
     year = {1999},
     volume = {6},
     doi = {10.37236/1444},
     zbl = {0915.05084},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1444/}
}
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%A S.-T. Yau
%T Coverings, heat kernels and spanning trees
%J The electronic journal of combinatorics
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Fan Chung; S.-T. Yau. Coverings, heat kernels and spanning trees. The electronic journal of combinatorics, Tome 6 (1999). doi: 10.37236/1444

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