2014/11/19 by Paul Horn, Horn, Paul, Yong Lin +6 · 4 citations
Mathematics · #Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations #math.CO #math.DG #math.MG
paper · pdf · doi:10.48550/arxiv.1411.5087
45 pages. arXiv admin note: text overlap with arXiv:0801.0812, arXiv:0911.1819 by other authors
openalex publication_date 2014/11/19 · arxiv created 2015/12/06 · arxiv updated 2015/12/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.