Global gradient estimate on graph and its applications

TL;DR

This paper derives a new global gradient estimate for positive solutions to the heat equation on graphs, enabling bounds on heat kernels and comparable results to Li-Yau inequalities, advancing analysis on graph structures.

Contribution

It introduces a novel global gradient estimate for positive functions on graphs, independent of previous estimates, with applications to heat kernel bounds and spectral analysis.

Findings

Provides upper and lower bounds for heat kernels on graphs

Establishes a new gradient estimate independent of prior work

Enables similar results to Li-Yau inequalities for eigenvalues and heat kernels

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.