Lei–Bai conjecture on 5-regular Lin–Lu–Yau Ricci-flat graphs

Conjecture 1 (Lei and Bai). If $G$ is a Ricci-flat $5$ -regular graph, then $G$ is either isomorphic to $RF_{72}^{5}$ or is a Cartesian product of the Petersen graph, the Triplex graph, or the dodecahedral graph with a cycle of length at least $6$ , or the infinite path. See Conjecture 1 in oss.linstitute.net.

Progress summary

Solved

A new unrefereed preprint claims an infinite family disproves the proposed classification, so the conjecture is not settled as a theorem.

The Lei–Bai conjecture proposes that every Ricci-flat $5$ -regular graph is either $RF_5^{72}$ or a Cartesian product of a Ricci-flat cubic graph and a cycle. Lei and Bai established only the symmetric case, while the broader classification remained conjectural.

portal.mardi4nfdi.de · oss.linstitute.net

Known results

Lei and Bai (2023): every symmetric Ricci-flat $5$ -regular graph is isomorphic to $RF_5^{72}$ .

portal.mardi4nfdi.de · oss.linstitute.net

August 2026 counterexample claim

Guangfu Wang, Wensheng Sun, and Yujun Yang claim an infinite family of $5$ -regular Lin–Lu–Yau Ricci-flat graphs outside the proposed classification, refuting its general form. The construction and curvature verification appear in an unrefereed preprint and have not been independently verified.

Current status (as of August 2026): The symmetric classification is proved, while the general Lei–Bai classification is claimed false by an unverified preprint and therefore remains unsettled.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

On the Lei--Bai conjecture on 5-regular Lin--Lu--Yau Ricci-flat graphs — arXiv — Guangfu Wang, Wensheng Sun, Yujun Yang

discrete-geometry graph-theory

4 days ago