Published September 15, 2026
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Curvature-Distortion Numbers of Graphs: Prescribed Lin--Lu--Yau Curvature Lower Bounds
Description
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.
Files
Curvature_Distortion_Numbers_of_Graphs__Prescribed_Lin__Lu__Yau_Curvature_Lower_Bounds.pdf
Additional details
Related works
- Continues
- Preprint: arXiv:2609.12125 (arXiv)