We introduce a modified heat equation which coincides with the the logarithmic heat equation on manifolds but not on graphs due to the lack of the chain rule. For the modified heat equation, we prove a gradient decay and Li-Yau inequality implying Harnack inequality. Although the modified heat equation does not coincide with the heat semigroup, we can give upper and lower bounds by which we can give a volume doubling constant depending only on the dimension and the vertex degree.

海报