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

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 \[\DN_{\LLY}^{k}(G):=\inf\bigl\{\Dist(w):\kappa_{\LLY}^{w}(e)\ge k\text{ for every }e\in E(G)\bigr\},\] 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 finite trees, the optimization problem has a sharp structure. If $T$ has at least two edges, then finite distortion is possible precisely for \[0\le k\le\frac{2}{|E(T)|}.\] For every such $k$, after normalizing the smallest edge weight to $1$, the optimal weight is unique. These optimal weights form a continuous and monotone family as the prescribed curvature increases from $0$ to its maximal possible value. They therefore determine a natural curvature--distortion profile joining the nonnegative-curvature problem to the maximal attainable uniform curvature. 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.