Global gradient estimate on graph and its applications

Abstract

Continuing our previous work (arXiv:1509.07981v1), we derive another global gradient estimate for positive functions, particularly for positive solutions to the heat equation on finite or locally finite graphs. In general, the gradient estimate in the present paper is independent of our previous one. As applications, it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs. These global gradient estimates can be compared with the Li-Yau inequality on graphs contributed by Bauer, Horn, Lin, Lipper, Mangoubi and Yau (J. Differential Geom. 99 (2015) 359-409). In many topics, such as eigenvalue estimate and heat kernel estimate (not including the Liouville type theorems), replacing the Li-Yau inequality by the global gradient estimate, we can get similar results.