Gradient Estimates and CDpψ (m, K) Curvature for the p-Laplacian on Weighted Graphs

In this paper, we study the Li-Yau's gradient estimate for the p-Laplacian on weighted graphs. For p >= 2, under the condition of CDp psi(m, K) curvature, we derive a more general type of Li-Yau's gradient estimate for positive solutions to the p-Laplacian heat equation on finite graphs or locally finite graphs with bounded weighted vertex degree. It is parallel to a result of Kotschwar and Ni on manifolds and an outcome of Wang on smooth metric measure spaces. In particular, when p = 2, our estimate deduces to L & uuml; and Wang's result on graphs with the CD psi(m, K) curvature. As an application of our main result, we derive the corresponding Harnack inequality.