Volume doubling, Poincar\'e inequality and Guassian heat kernel estimate for nonnegative curvature graphs

TL;DR

This paper establishes fundamental geometric and analytic properties of graphs with non-negative curvature, including volume doubling, Gaussian heat kernel estimates, and finiteness of harmonic functions, extending classical Riemannian results to graph settings.

Contribution

It introduces curvature-dimension inequalities for graphs and proves key inequalities and properties analogous to Riemannian geometry, such as volume doubling and heat kernel estimates.

Findings

Graphs with non-negative curvature satisfy volume doubling.

Gaussian heat kernel estimates hold for such graphs.

Finite-dimensionality of harmonic functions with polynomial growth is established.

Abstract

By studying the heat semigroup, we prove Li-Yau type estimates for bounded and positive solutions of the heat equation on graphs, under the assumption of the curvature-dimension inequality $CDE'(n,0)$, which can be consider as a notion of curvature for graphs. Furthermore, we derive that if a graph has non-negative curvature then it has the volume doubling property, from this we can prove the Gaussian estimate for heat kernel, and then Poincar\'e inequality and Harnack inequality. As a consequence, we obtain that the dimension of space of harmonic functions on graphs with polynomial growth is finite, which original is a conjecture of Yau on Riemannian manifold proved by Colding and Minicozzi. Under the assumption of positive curvature on graphs, we derive the Bonnet-Myers type theorem that the diameter of graphs is finite and bounded above in terms of the positive curvature by proving some Log Sobolev inequalities.