Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for non-negatively curved graphs

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 Poincaré 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.

Ollivier Ricci curvature for general graph Laplacians: Heat equation, Laplacian comparison, non-explosion and diameter bounds

“…Diameter bounds under a positive average Ollivier curvature can be found in [Pae12]. Diameter bounds under uniformly positive Bakry-Emery curvature are proven in [FS18;Hor+14;LMP16]. In this article we show that if the vertex degree is bounded and the curvature decays not faster than 1/R, then the graph is finite (see Theorem 4.15 and Corollary 4.16).…”

Log-Sobolev Inequalities on Graphs With Positive Curvature

“…and since ∫︀^2 µ = ∫︀ 2 µ − (︀∫︀ µ )︀ 2 for the probability measure µ (the finiteness of measure is true when the graph satisfies ′ ( , ) with > 0, see [8]), it remains to use the Poincaré inequality ( ′ ), the conclusion is therefore established.…”

“…Now we introduce the notion of the ′ inequality on graphs from [8]. First we need to recall the definition of two bilinear forms associated to the µ-Laplacian.…”

“…(see [8,Proposition 7.5]). Then, by Holdër inequality and combining with (2.2), we obtain for some 0 > 0,…”

“…Before the next step, we first show two theorems from [8] and [2]. It will be useful to prove the graph also satisfy a Poincaré inequality when the curvature of graph is bounded by a positive number.…”