An extremal theorem for positive curvature of graphs

Abstract: We prove an extremal theorem for positive Ollivier/Lin--Lu--Yau curvature: every graph of order (n\geq 8) with more than [ T(n)=\frac{n^2-3n}{2}-\left\lceil\frac{n}{2}\right\rceil+2 ] edges has positive Ollivier/Lin--Lu--Yau curvature, and this threshold is optimal. Moreover, for even $n\geq 12$, there exists a unique graph with $T(n)$ edges that has an edge with non-positive curvature. For $n=8,10$ and odd $n\geq 9$, the extremal graphs are not unique. This suggests a new class of extremal graph-theoretic problems arising from discrete curvature notions.

• The paper establishes the exact edge threshold T(n) for graphs of order n ≥ 8 to guarantee strictly positive Ollivier/LLY curvature.
• It employs optimal transport theory and combinatorial methods to translate curvature constraints into precise edge counts and graph structures.
• Unique extremal constructions for even n ≥ 12 and multiple configurations for odd and small even n highlight implications for network design and discrete geometric analysis.

Extremal Thresholds for Positive Ollivier/Lin–Lu–Yau Curvature in Graphs

Introduction and Context

The study addresses an extremal question at the intersection of discrete geometry and graph theory: for a fixed number of vertices $n$, what is the sharp threshold on the number of edges ensuring that every edge in a graph admits strictly positive Ollivier/Lin–Lu–Yau (LLY) curvature? Building upon discrete analogs of Ricci curvature—especially the Ollivier and LLY formulations—this work identifies not only explicit edge thresholds but also characterizes extremal graphs at and near the threshold, elucidating the dependencies of positive curvature on combinatorial density.

Formal Statement of Results

The central theorem (Theorem 1) establishes that for any graph of order $n \geq 8$, if the number of edges exceeds

$$T(n) = \frac{n^2 - 3n}{2} - \left\lceil \frac{n}{2} \right\rceil + 2,$$

then the graph necessarily has positive LLY curvature on all edges. For even $n \geq 12$, the extremal graph with exactly $T(n)$ edges and exactly one edge with non-positive curvature is unique. For $n = 8, 10$ and all odd $n \geq 9$, the extremal graphs at the threshold are not unique, and distinct constructions are given. The proof further demonstrates that both the lower bound on $n$ and the formula for $T(n)$ cannot be improved.

Discrete Ricci Curvature in Graphs

Ollivier curvature $\kappa_p(x,y)$ is formulated via the $n \geq 8$0-Wasserstein distance between $n \geq 8$1-lazy random walk measures $n \geq 8$2 on adjacent vertices $n \geq 8$3:

$n \geq 8$4

The Lin–Lu–Yau curvature $n \geq 8$5 is a scaling limit as $n \geq 8$6, encoding sensitivity to "idleness" in the random walk. Positive curvature intuitively enforces dense local neighborhoods; low-degree vertices and sparse edge sets tend to induce negative curvature locally.

Proof Outline and Extremal Construction

The proof employs optimal transport theory to characterize when $n \geq 8$7. The argument involves selecting a maximizing transport plan, then translating curvature non-positivity into restrictions on adjacencies and degrees. These combinatorial constraints, combined with global edge counting, are reduced to the resolution of a specific numerical inequality (Lemma 3.1), explored with case distinctions.

A pivotal insight is that the minimal complements for positivity correspond, in terms of forbidden subgraphs, to highly structured graphs (often featuring large cliques with a small number of edges removed). The extremal graphs are constructed by identifying small sets of edges which, if absent from the complete graph, yield an edge attaining zero (or negative) LLY curvature—see Figure 1:

For even $n \geq 8$8, the extremal graph is unique at the threshold $n \geq 8$9 and is precisely described. For odd $$T(n) = \frac{n^2 - 3n}{2} - \left\lceil \frac{n}{2} \right\rceil + 2,$$0 and small even $$T(n) = \frac{n^2 - 3n}{2} - \left\lceil \frac{n}{2} \right\rceil + 2,$$1, the boundary case admits multiple, non-isomorphic extremal constructions with explicit descriptions provided in the paper.

Implications and Connections to Extremal Graph Theory

The theorem refines our understanding of how discrete geometric constraints (here, positive Ricci curvature) translate to stringent edge density conditions. It invites a new class of extremal problems, in analogy with Turán-type or spectral extremal results, but driven by geometric (curvature) considerations instead of classical forbidden subgraphs or eigenvalue bounds.

The explicit identification of threshold functions and extremal configurations enables further bounding of other discrete geometric invariants—such as diameter, expansion, or spectral gap—on curved graphs. Practically, these results motivate new graph design criteria for network science where "geometric robustness" (positive curvature) is desirable.

Theorem 1's complementary statements, including in terms of forbidden subgraphs in the complement and edge counts in the complement, further enrich the combinatorial-geometric dictionary and provide sharper tools for analyzing both theoretical and applied networks.

Directions for Future Work

The authors suggest generalizations of the extremal question along several axes:

1. Thresholds for Strictly Larger Curvature: Determine, for any $$T(n) = \frac{n^2 - 3n}{2} - \left\lceil \frac{n}{2} \right\rceil + 2,$$2, the edge threshold $$T(n) = \frac{n^2 - 3n}{2} - \left\lceil \frac{n}{2} \right\rceil + 2,$$3 such that every graph with more than $$T(n) = \frac{n^2 - 3n}{2} - \left\lceil \frac{n}{2} \right\rceil + 2,$$4 edges satisfies $$T(n) = \frac{n^2 - 3n}{2} - \left\lceil \frac{n}{2} \right\rceil + 2,$$5 for all edges.
2. Defect Allowance: For any $$T(n) = \frac{n^2 - 3n}{2} - \left\lceil \frac{n}{2} \right\rceil + 2,$$6, determine the minimal edge count such that all but $$T(n) = \frac{n^2 - 3n}{2} - \left\lceil \frac{n}{2} \right\rceil + 2,$$7 edges have strictly positive LLY curvature.
3. Saturation Problems: Characterize the minimal graphs such that each addition of a non-edge creates strictly positive curvature everywhere.
4. Alternative Curvature Notions: Investigate analogous extremal problems for Bakry-Émery or resistance-based curvatures, which may have different combinatorial sensitivities.

These problems may yield new forms of "curvature-Turán" theorems and guide the development of discrete geometric analysis in network design, metric geometry, and eigenvalue optimization.

Conclusion

This work rigorously determines the precise edge density thresholds required for global positivity of Ollivier/Lin–Lu–Yau curvature in finite simple graphs and classifies the extremal graphs attaining these bounds. The results underline the close interplay between combinatorial structure and discrete geometric properties and open multiple new avenues for extremal, structural, and applied analysis of curvature in graphs.