Classify graphs attaining the sharp one-half curvature threshold

Classify all finite simple graphs for which the necessary equality conditions in the one-half curvature estimate are simultaneously attained, including the required support and coefficient conditions for a normalized circulation and the requirement that the compressed McShane extension minimize the Lin–Lu–Yau variational formula.

Background

The paper proves that any finite simple graph with nonzero first GLMY path homology has an edge of Lin–Lu–Yau curvature at most one-half, and that the five-cycle attains this threshold. The proof identifies necessary equality conditions at the two endpoints of an edge supporting a maximal circulation: the positive and negative supports must exhaust the incident edges, have equal cardinality, and have coefficients of absolute value one. Equality also requires the compressed McShane extension to be an optimizer in the variational characterization of curvature.

The paper explicitly leaves unresolved the classification of graphs in which all these conditions occur simultaneously. Such a classification would clarify the structure of extremizers for the sharp one-half theorem beyond the example of the five-cycle.

References

We do not attempt here to classify all graphs for which these necessary conditions are simultaneously attained.

— A Sharp Curvature Threshold for GLMY Path Homology  (2608.23187 - Bai et al., 24 Aug 2026) in Remark 2.14, Section 2 (following the proof of the sharp one-half vanishing theorem)

Determine $K_p{\mathrm{LLY}}$. Is $K_p{\mathrm{LLY}}=1/(2p)$ for every $p$? If so, classify the extremal graphs and determine whether $C_5{\square p}$ is rigid among extremizers.

— A Sharp Curvature Threshold for GLMY Path Homology  (2608.23187 - Bai et al., 24 Aug 2026) in Problem 4.4, Section 4.4 “An extremal curvature problem”