Extremal Lin--Lu--Yau Curvature: Graph Density, Girth, and Short Cycles
We consider the extremal-curvature problem of optimizing a uniform discrete-curvature lower bound over positive edge weights, and develop this problem here for Lin--Lu--Yau curvature. Let $G=(V,E)$ be a finite connected graph, and let $w:E\to(0,\infty)$ be a positive edge weight. In the fixed-combinatorial-distance weighted Lin--Lu--Yau model, write \[ \kappa_{\LLY}^w(G):=\min_{e\in E}\kappa_{\LLY}^w(e) \] and define the extremal Lin--Lu--Yau curvature \[ \Kmax(G):=\sup_{w>0}\kappa_{\LLY}^w(G). \] For graphs of girth at least $6$ we determine this invariant exactly: \[ \Kmax(G)=\frac{4}{\mad(G)}-2, \] where $\mad(G)$ is the maximum average degree. Equivalently, \[ \Kmax(G) =\min_{\substack{H\subseteq G\text{ connected}\\E(H)\ne\varnothing}} \frac{2(1-\beta(H))}{|E(H)|}, \] where $\beta(H)=|E(H)|-|V(H)|+1$ is the cycle rank of the connected graph $H$. Thus, in the high-girth regime, the invariant is a normalized Euler-characteristic density. We characterize attainment in terms of the classical notion of strict balancedness and show that maximizing sequences concentrate, in a precise normalized-incidence sense, on proper densest cores when the supremum is not attained. For arbitrary finite graphs we isolate the contribution of short cycles by a nonnegative surplus, which vanishes exactly on edges contained in no cycle of length $3$, $4$, or $5$. For edges contained in no triangle, this surplus is the value of an explicit local fractional matching problem. This yields the sharp hierarchy \[ \Kmax(G)\le 4-\ell+\frac{\ell-2}{\mad(G)}, \qquad \girth(G)\ge\ell,\quad \ell\in\{3,4,5,6\}, \] with equality for every finite connected graph when $\ell=6$. We also prove that the attainment is rigid.