Geodesic
A discrete analogue of the harmonic morphism and green kernelcomparison theorems
Glasgow mathematical journal, Tome 42 (2000) no. 3, pp. 319-334

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We give a discrete analogueof the harmonic morphism between two Riemannian manifolds. Roughly speaking, this is a mapping betweentwo graphs preserving local harmonic functions. We characterize harmonic morphisms in terms ofhorizontal conformality. Many examples including coverings, non-complete extended p-sums andcollapsings are given. Introducing the horizontal and vertical Laplacians, the Green kernel estimatesare obtained for the harmonic morphism. As applications, a general and sharp estimate of the Greenkernel for an infinite tree is obtained. 
Urakawa, Hajime. A discrete analogue of the harmonic morphism and green kernelcomparison theorems. Glasgow mathematical journal, Tome 42 (2000) no. 3, pp. 319-334. doi: 10.1017/S0017089500030019
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