Heat Kernel and Green Kernel Comparison Theorems for Infinite Graphs

For an infinite graph, general lower and upper estimates of the Green kernel and the heat kernel are given. The estimates are optimal in the case of the homogeneous regular trees. As their applications, solvability of Dirichlet problem for the end compactification is shown and the sharp estimates of several infinite graphs including the distance regular graphs and the free products of finite complete graphs are given.