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Antiferromagnetic models as clique-minimisers

From papers

Let GG be a graph, let dv≔deg⁡G(v)d_v\coloneqq\deg_G(v), and let HH be an edge-weighted graph, possibly with loops. Define the weighted homomorphism count by

hom⁡(G,H)≔∑ϕ:V(G)→V(H)∏uv∈E(G)H(ϕ(u),ϕ(v)),\hom(G,H)\coloneqq\sum_{\phi:V(G)\rightarrow V(H)}\prod_{uv\in E(G)}H(\phi(u),\phi(v)),

where H(a,b)H(a,b) is the edge weight between aa and bb. Call HH antiferromagnetic if its symmetric edge-weight matrix has exactly one positive eigenvalue, counting multiplicity. The antiferromagnetic clique-minimiser conjecture. For every graph GG and every antiferromagnetic edge-weighted graph HH,

hom⁡(G,H)≥∏v∈V(G)hom⁡(Kdv+1,H)1dv+1.\hom(G,H)\geq\prod_{v\in V(G)}\hom(K_{d_v+1},H)^{\frac{1}{d_v+1}}.

This conjecture would extend the lower bound for multivariate independence polynomials and its two-spin corollary to all antiferromagnetic homomorphism models. It is established in the paper for the relevant two-spin models, but the general antiferromagnetic case remains open.

Progress summary

Partially solved

The broad conjecture remains open, with a February 2026 paper proving only important special cases.

The conjecture asserts a universal lower bound for weighted homomorphism counts into every antiferromagnetic target graph. Its general form is explicitly left as Conjecture 1.31.3 in the latest retrieved paper.

Known results

  • The multivariate independence-polynomial and two-spin cases follow from Theorems 1.11.1 and 1.21.2.
  • The bound is proved for targets H=KqH=K_q and H=K3∘H=K_3^\circ.

February 2026 partial progress

The paper “Lower bounds for multivariate independence polynomials and their generalisations” develops further vertex-weighted results for H=Kq+1∘H=K_{q+1}^\circ and describes them as partial progress toward Conjecture 1.31.3. It also says the authors independently rewrote selected arguments whose statements were suggested by an unnamed model; this is not a model-produced proof of the conjecture.

Current status (as of August 2026): The general antiferromagnetic clique-minimiser conjecture remains open; the February 2026 paper establishes only the stated special cases and partial results.

Sources
Sources & referencesView supporting material

Primary source

Joonkyung Lee and Jaehyeon Seo, “Lower bounds for multivariate independence polynomials and their generalisations”, arXiv:2602.02450 (2026).

antiferromagnetic-modelsextremal-graph-theorygraph-homomorphisms

Solutions 1

Proof

Joonkyung Lee and I just uploaded a proof to arXiv: see https://arxiv.org/abs/2608.17920 .

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