A rigidity theorem for edge-connectivity under non-negative Bakry--Émery curvature

Abstract

The edge-connectivity $κ(G)$ of a graph $G$ is the minimum number of edges whose removal disconnects $G$, and it is bounded above by the minimum degree $δ(G)$. We classify all connected locally finite simple graphs with nonnegative Bakry--Émery curvature satisfying $κ(G)<δ(G)$. All such graphs are infinite. Consequently, every finite connected simple graph with nonnegative Bakry--Émery curvature satisfies $κ(G)=δ(G)$, confirming a conjecture of Chen, Koolen, and Liu.