Graphs with large girth and nonnegative curvature dimension condition (Q2322320)

Geometric analysts have found so-called curvature dimension conditions that generalize to metric spaces the geometry of lower bounds on Ricci curvature, which can a priori be defined for Riemannian manifolds only. In particular, Riemannian manifolds of dimension at most \(n\) and Ricci curvature at least \(K\) satisfy the curvature dimension condition \(\mathrm{CD}(K,n)\). The classical approach by Bakry and Émery uses a Dirichlet form, and the generator \(\Delta\) of the associated Markov semigroup, to define a curvature dimension condition. The paper under review considers the analogous definitions for graphs with the role of \(\Delta\) taken by the graph Laplacian, and classifies all graphs of ``nonnegative Ricci curvature'', i.e., satisfying \(\mathrm{CD}(0,\infty)\) in this sense. Let \((V,E)\) be a graph and \(\Delta f(x)=\frac{1}{\deg(x)}\sum_{y\sim x}(f(y)-f(x))\) its normalized Laplacian. As in the classical case the ``square of the field'' operator is defined by \(\Gamma(f,g)=\frac{1}{2}(\Delta(fg)-f\Delta g-g\Delta f)\) and the iterated gradient by \(\Gamma_2(f)=\frac{1}{2}\Delta\Gamma(f)-\Gamma(f,\Delta f)\), and a graph is said to satisfy \(\mathrm{CD}(K,n)\) if \(\Gamma_2(f)(x)\ge \frac{1}{n}(\Delta f(x))^2+K\Gamma(f)(x)\) for all \(x\in V\). In particular, \(\mathrm{CD}(0,\infty)\) holds if there is a positive lower bound on \(\frac{\Gamma_2(f)(x)}{(\Delta f(x))^2}\). The main result of the paper under review is to classify all graphs satisfying the \(\mathrm{CD}(0,\infty)\) condition among the graphs of girths at least 5, i.e., among graphs not containg a 3- or 4-gon. The graphs satisfying these conditions are the following: the path graphs \(P_k\) for \(k\ge 1\), the cycle graphs \(C_n\) for \(n\ge 5\), the infinite line \(P_{\mathbb Z}\), the infinite half-line \(P_{\mathbb N}\), the star graphs \(St_n\) for \(n\ge 3\) or \(St_3^i\) for \(1\le i\le 3\) (the star graph \(St_3\) with \(i\) edges added).