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{\bf Clara E. Brasseur, Ryan E. Grady and Stratos Prassidis}
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{\bf Coverings, Laplacians, and Heat Kernels of Directed Graphs}
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Combinatorial covers of graphs were defined by Chung and Yau. Their
main feature is that the spectra of the Combinatorial Laplacian of the
base and the total space are related.  We extend their definition to
directed graphs. As an application, we compute the spectrum of the
Combinatorial Laplacian of the homesick random walk $RW_{\mu}$ on the
line. Using this calculation, we show that the heat kernel on the
weighted line can be computed from the heat kernel of 
`$(1 + 1/\mu)$-regular' tree.



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