Extremal Lin--Lu--Yau Curvature: Graph Density, Girth, and Short Cycles

Abstract

We develop a variational framework for extremal Lin--Lu--Yau curvature under positive edge reweighting with the combinatorial metric fixed. For a connected locally finite graph $G$, define \[ \Kmax(G) := \sup_{w>0}\inf_{e\in E(G)}\kappa_{\LLY}^w(e). \] This invariant relates edgewise discrete curvature to global combinatorial and topological data. In the high-girth regime, we prove for every connected locally finite graph the exact identity \[ \Kmax(G)=\frac4{\mad(G)}-2, \] with the convention $1/\infty=0$. The formula is also equivalent to a normalized Euler-characteristic density: \[ \Kmax(G) = \inf_{\substack{H\subseteq G\text{ finite}\\E(H)\ne\varnothing}} \frac{2(c(H)-\beta(H))}{|E(H)|}. \] For arbitrary locally finite graphs, a nonnegative short-cycle surplus yields girth-dependent density bounds, with rigidity results in the finite case. We also classify the locally finite graphs of girth at least $6$ admitting a positive edge weight with nonnegative curvature on every edge: they are precisely the trees and the finite cycles.