A gradient estimate for positive functions on graphs

Abstract: 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.