Lorentzian characterization of clique source graphs

Determine whether, for a connected graph G, the vertex-homogeneous partition function Z_G of every antiferromagnetic model H is Lorentzian if and only if G is a clique.

Background

The paper proves that clique partition functions of entrywise-positive antiferromagnetic models are Lorentzian. The concluding remarks discuss a broader conjecture concerning which connected source graphs preserve this Lorentzian property for every antiferromagnetic target model.

The conjecture would characterize cliques among connected source graphs: cliques have the Lorentzian property established in the paper, while the unresolved direction asks whether no other connected source graphs have it universally.

References

We note that it was conjectured in Conjecture~7.1 that, for a connected graph $G$, the vertex-homogeneous partition function $Z_G$ of $H$ is Lorentzian for every antiferromagnetic model $H$ if and only if $G$ is a clique. Thus, this conjecture predicts that the Lorentzian property established in \Cref{H-antiferromagnetic-Z-Lorentzian} characterizes cliques among connected source graphs.

— Antiferromagnetic models are clique-minimizing  (2608.17920 - Lee et al., 18 Aug 2026) in Section 6, “Concluding remarks”