Connectivity versus Lin-Lu-Yau curvature
Kaizhe Chen, Shiping Liu, Zhe You
TL;DR
This work investigates the relationship between graph connectivity and Lin-Lu-Yau curvature, introducing curvature-based lower bounds on connectivity and identifying conditions under which curvature dictates edge-connectivity. It proves a key bound $k(G)\ge \delta(G)\kappa_{LLY}^{(2)}(G)$ and shows that positive curvature on edges implies $k'(G)=\delta(G)$, while large connectivity enforces positive curvature in a quantified way. The results are shown to be sharp via constructions such as deletions of matchings from complete graphs and Cartesian-type joins, with extensions to amply regular graphs yielding explicit lower bounds for both connectivity and edge-connectivity. Applications to amply regular graphs demonstrate practical bounds, and the work also discusses implications for infinite locally finite graphs. Overall, the paper deepens the link between discrete Ricci-type curvature and classical connectivity measures, providing tools for both theoretical and applied graph analysis.
Abstract
We explore the interaction between connectivity and Lin-Lu-Yau curvature of graphs systematically. The intuition is that connected graphs with large Lin-Lu-Yau curvature also have large connectivity, and vice versa. We prove that the connectivity of a connected graph is lower bounded by the product of its minimum degree and its Lin-Lu-Yau curvature. On the other hand, if the connectivity of a graph $G$ on $n$ vertices is at least $\frac{n-1}{2}$, then $G$ has positive Lin-Lu-Yau curvature. Moreover, the bound $\frac{n-1}{2}$ here is optimal. Furthermore, we prove that the edge-connectivity is equal to the minimum vertex degree for any connected graph with positive Lin-Lu-Yau curvature. As applications, we estimate or determine the connectivity and edge-connectivity of an amply regular graph with parameters $(d,α,β)$ such that $1\neq β\geq α$.