An extremal theorem for positive curvature of graphs

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