  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.
