Gradient estimates for the heat equation on graphs

Let G(V, E) be an infinite (locally finite) graph which satisfies C DE(m, -K) condition for some m > 0, K > 0. In this paper we mainly establish a generalized gradient estimate for positive solutions to the following heat equation (partial derivative(t )- Delta(mu))u = 0, this gradient estimate includes Davies' estimate, Hamilton's estimate, Bakry-Qian's estimate and Li-Xu's estimate, these four estimates for positive solutions to the linear heat equation had been established on complete manifolds with Ricci curvature bounded from below by a negative number. When t SE arrow 0, the dominant terms of Hamilton, Bakry-Qian and Li-Xu's estimates are consistent with the corresponding term on the case that K = 0. We can also derive the Harnack inequalities from the gradient estimates, and then obtain the heat kernel estimates.