  We derive a gradient estimate for positive functions, in particular for
positive solutions to the heat equation, on finite or locally finite graphs.
Unlike the well known Li-Yau estimate, which is based on the maximum principle,
our estimate follows from the graph structure of the gradient form and the
Laplacian operator. Though our assumption on graphs is slightly stronger than
that of Bauer, Horn, Lin, Lippner, Mangoubi, and Yau (J. Differential Geom. 99
(2015) 359-405), our estimate can be easily applied to nonlinear differential
equations, as well as differential inequalities.
  As applications, we estimate the greatest lower bound of Cheng's eigenvalue
and an upper bound of the minimal heat kernel, which is recently studied by
Bauer, Hua and Yau (Preprint, 2015) by the Li-Yau estimate. Moreover,
generalizing an earlier result of Lin and Yau (Math. Res. Lett. 17 (2010)
343-356), we derive a lower bound of nonzero eigenvalues by our gradient
estimate.
