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arXiv · 2607.02297

An extremal theorem for positive curvature of graphs

Kaizhe Chen·Shiping Liu·Zhe You
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Abstract

We prove an extremal theorem for positive Ollivier/Lin--Lu--Yau curvature: every graph of order \(n\geq 8\) with more than \[ T(n)=\frac{n^2-3n}{2}-\left\lceil\frac{n}{2}\right\rceil+2 \] edges has positive Ollivier/Lin--Lu--Yau curvature, and this threshold is optimal. Moreover, for even $n\geq 12$, there exists a unique graph with $T(n)$ edges that has an edge with non-positive curvature. For $n=8,10$ and odd $n\geq 9$, the extremal graphs are not unique. This suggests a new class of extremal graph-theoretic problems arising from discrete curvature notions.

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Kaizhe Chen, Shiping Liu, Zhe You. 2026-07-02. An extremal theorem for positive curvature of graphs. https://arxiv.org/abs/2607.02297

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