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

We prove that every connected simple graph of bounded degree satisfying the classical dimension-free Bakry–Émery condition $\mathrm{C}\mathrm{D}(0,\mathrm{\infty })$ for the unnormalised Laplacian is volume doubling and supports, at all integer graph scales, a scale-invariant $L^2$-Poincaré 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 Münch and extended to infinite weighted graphs by Pajot and Russ: point-mass consequences of ${\mathrm{\Gamma }}_2\ge 0$ and positive-resolvent smoothing replace any global $\mathrm{C}\mathrm{D}(0,n)$ reduction, while diffusive exit-time control and finite-volume localisation yield the Poincaré inequality.