Nonnegative Bakry--Émery Curvature on Bounded-Degree Graphs Implies Volume Doubling and Poincaré Inequalities

Qi Guo; Xueping Huang; Yi C. Huang

Nonnegative Bakry--Émery Curvature on Bounded-Degree Graphs Implies Volume Doubling and Poincaré Inequalities

Differential Geometry 2026-07-17 v1 Combinatorics Probability

Authors: Qi Guo , Xueping Huang , Yi C. Huang

Abstract

We prove that every connected simple graph of bounded degree satisfying the classical dimension-free Bakry--\'Emery condition $\mathrm{CD}(0,\infty)$ for the unnormalised Laplacian is volume doubling and supports, at all integer graph scales, a scale-invariant $L^2$ -Poincar\'e inequality with dilation two, with constants depending only on the maximum degree. This settles the polynomial-growth conjecture of Cushing, Liu, and Peyerimhoff in a stronger form. The main novelty is a dimension-free adaptation of the graph-theoretic modified nonlinear heat-flow method introduced by M\"unch and extended to infinite weighted graphs by Pajot and Russ: point-mass consequences of $\Gamma_2\geq0$ and positive-resolvent smoothing replace any global $\mathrm{CD}(0,n)$ reduction, while diffusive exit-time control and finite-volume localisation yield the Poincar\'e inequality.

Keywords

bakry-emery curvature bounded-degree graphs volume doubling poincare inequality nonlinear heat flow graph laplacian

Cite

@article{arxiv.2607.15522,
  title  = {Nonnegative Bakry--Émery Curvature on Bounded-Degree Graphs Implies Volume Doubling and Poincaré Inequalities},
  author = {Qi Guo and Xueping Huang and Yi C. Huang},
  journal= {arXiv preprint arXiv:2607.15522},
  year   = {2026}
}