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Combinatorial covers of graphs were dened 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 denition to directed graphs. As an application, we compute the spectrum of the Combinatorial Laplacian of the homesick random walk RW 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= )-regular' tree.

Coverings, Laplacians, and Heat Kernels of Directed Graphs