Heat Kernel and Monotonicity Inequalities on the Graph

Let G = (V, E) be a connected locally finite graph with the CD psi condition for some C-1, concave function psi : (0, +infinity)-* R. In this paper, based on the gradient estimate and Harnack inequality for the positive solution to the heat equation on G established by Munch, we get the heat kernel estimate. We also derive a logarithmic inequality of the heat kernel by an observation of the short time behavior of the heat kernel. We also define the Fisher information, the Shannon entropy and the W-functional, and establish monotone inequalities for these functionals along the heat equation on G.