Curvature-Distortion Numbers of Graphs: Prescribed Lin--Lu--Yau Curvature Lower Bounds

Source abstract

We study an optimization problem for weighted Lin--Lu--Yau curvature. Given a graph $G=(V,E)$ and a prescribed lower curvature bound $k\ge0$, we ask for the smallest multiplicative distortion \[ \Dist(w):=\frac{\sup_{e\in E}w_e}{\inf_{e\in E}w_e} \] of the edge weights needed to make every edge have curvature at least $k$. This leads to the family \[ \DN_{\LLY}^{k}(G) := \inf\bigl\{ \Dist(w): κ_{\LLY}^{w}(e)\ge k \text{ for every }e\in E(G) \bigr\}, \] which extends the nonnegative-curvature invariant introduced in our preceding paper \cite{Xia2026InvariantI}. The edge weights change the transition probabilities, while the combinatorial graph distance remains fixed. Positive prescribed curvature leads to a fundamental distinction from the case $k=0$. If $G$ is a connected locally finite infinite graph and $k>0$, then \[ \DN_{\LLY}^{k}(G)=\infty. \] Thus no finite multiplicative distortion of the edge weights can impose a uniform positive curvature lower bound on such a graph. For a finite tree $T$ with $m$ edges, such weights exist exactly when $k\le2/m$. We prove that the minimum distortion is attained by a weight unique up to scaling, and that the normalized minimizer and its distortion depend continuously and nondecreasingly on $k$. We then extend the finite-tree theory to general connected graphs. As an application, we characterize the graphs of girth at least six for which finite $k$-distortion is possible.