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arXiv · 2504.14352

Connectivity versus Lin-Lu-Yau curvature

Kaizhe Chen·Shiping Liu·Zhe You
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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 $1\neq β\geq α$.

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Kaizhe Chen, Shiping Liu, Zhe You. 2025-04-19. Connectivity versus Lin-Lu-Yau curvature. https://arxiv.org/abs/2504.14352

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