Antiferromagnetic models are clique-minimizing

Abstract: An edge-weighted graph $H$, possibly with loops, is antiferromagnetic if its adjacency matrix is entrywise nonnegative and has at most one positive eigenvalue, counted with multiplicity. We show that, for any graph $G$ with $d_v:=\operatorname{deg}<em>G(v)$, $$\operatorname{hom}(G,H) \ge \prod</em>{v\in V(G)} \operatorname{hom}(K_{d_v+1},H)<sup>{\frac{1}{d_v+1}},$$ whenever $H$ is antiferromagnetic. In fact, we prove a vertex-inhomogeneous strengthening of this inequality, allowing a different fugacity vector at each vertex of $G$. This gives a common generalization of the lower-bound inequalities of Sah, Sawhney, Stoner, and Zhao for independent sets, of Csikvári for $q$-colorings, and of the authors for semiproper colorings with at most two proper colors. Furthermore, it confirms recent conjectures of the authors and of Davies and LeBlanc. A key ingredient, of independent interest, is a strengthening of the delete-one form of Shearer's inequality for Lorentzian measures, which provides a new approach to graph homomorphism inequalities.

• The paper proves that every entrywise-nonnegative interaction matrix with at most one positive eigenvalue minimizes normalized homomorphism counts on complete graphs for arbitrary degree sequences.
• The vertex-inhomogeneous inequality remains sharp for disjoint unions of cliques, covering weighted independent sets, list colorings, list homomorphisms, and spin systems with varying external fields.
• The proof combines Lorentzian polynomial theory with relative entropy contraction for exchangeable Lorentzian laws, while leaving biclique maximization and a full characterization of clique-minimizing models open.

The clique-minimizing phenomenon and its scope

Complete graphs recur as extremizers in counting problems on graphs. Cutler and Radcliffe proved that the number of independent sets of a $d$-regular graph, normalized by taking $1/v(G)$-th powers, is minimized by $K_{d+1}$; Csikvári proved the analogous statement for proper $q$-colorings; subsequent work extended such inequalities to independence polynomials with arbitrary degree sequences, to antiferromagnetic Ising models on cubic graphs, and to semiproper colorings. These results were proved by model-specific arguments, and no general criterion explained when complete graphs should be minimizers. The paper under review resolves this question by identifying a spectral condition: an edge-weighted graph (model) $H$ on a spin set $I$, given by a symmetric entrywise-nonnegative matrix, is antiferromagnetic if it has at most one positive eigenvalue counted with multiplicity. The main theorem asserts that every antiferromagnetic model is clique-minimizing: for every graph $G$ with degrees $d_v$,

$$\hom(G,H)\ \ge\ \prod_{v\in V(G)} \hom(K_{d_v+1},H)^{1/(d_v+1)}.$$

For $d$-regular $1/v(G)$0 this reads $1/v(G)$1. The class of antiferromagnetic models contains the hard-core model $1/v(G)$2, complete graphs $1/v(G)$3, looped complete graphs $1/v(G)$4 encoding semiproper colorings, and antiferromagnetic Ising models with edge activity $1/v(G)$5. The theorem therefore recovers the known inequalities for independent sets and colorings, extends the coloring result to arbitrary degree sequences, and settles both the authors' earlier conjecture that all antiferromagnetic models are clique-minimizing and the Davies–LeBlanc conjecture on antiferromagnetic Ising models — a conjecture its proposers described as "somewhat bold."

Vertex-inhomogeneous strengthening and its sharpness

The proof establishes a strictly stronger statement in which each vertex $1/v(G)$6 of $1/v(G)$7 carries its own fugacity vector $1/v(G)$8. Writing $1/v(G)$9 for the corresponding vertex-inhomogeneous partition function and $K_{d+1}$0 for the clique partition function, the inequality reads

$K_{d+1}$1

Equality holds whenever $K_{d+1}$2 is a disjoint union of cliques with fugacity vectors constant on components, so the bound is sharp. The inhomogeneous formulation captures multivariate independence polynomials, list-coloring and weighted list-homomorphism problems, spin systems with vertex-dependent external fields, and one-vertex marginals central to the occupancy method. It also settles the authors' earlier conjecture on vertex-inhomogeneous inequalities for $K_{d+1}$3 and $K_{d+1}$4.

Notably, the inhomogeneous inequality characterizes antiferromagnetism: applying it to $K_{d+1}$5 with arbitrary fugacity vectors $K_{d+1}$6 yields $K_{d+1}$7, which by a theorem of Choe et al. forces $K_{d+1}$8 to be antiferromagnetic. The homogeneous clique-minimizing inequality, by contrast, does not characterize the class — the concluding section exhibits clique-minimizing models that are not antiferromagnetic, such as tensor products of antiferromagnetic models and a triangle with one pendant edge.

Position within homomorphism inequalities

The result completes the lower half of an expected dichotomy. Sah, Sawhney, Stoner, and Zhao showed that every ferromagnetic (positive-semidefinite) model is clique-maximizing: the same clique expression becomes an upper bound. Positive semidefiniteness and antiferromagnetism thus force the identical quantity to be an upper and a lower bound, respectively. For antiferromagnetic models the conjectured global extremal picture pairs clique lower bounds with biclique upper bounds; the biclique-maximizing conjecture of Sah et al. remains open. The paper's contribution is the lower bound in full generality, for all antiferromagnetic models and arbitrary degree sequences.

Methodology: Lorentzian polynomials and relative entropy contraction

Two algebraic and probabilistic ingredients drive the proof. First, the clique partition function $K_{d+1}$9 of an entrywise-positive antiferromagnetic model is Lorentzian in the sense of Brändén–Huh; in degree two the Hessian condition for Lorentzian polynomials coincides exactly with the antiferromagnetic spectral condition. Consequently $q$0 is concave, supplying the tangent-plane inequalities that anchor the induction. Second, the paper proves a strengthening of the delete-one case of Shearer's (equivalently, Han's) inequality for exchangeable Lorentzian laws — laws whose count-generating polynomial is Lorentzian, a multiset analogue of the Lorentzian probability measures of Brändén–Huh. The key tool is a one-coordinate relative entropy contraction, a multiset analogue of entropic independence of Anari–Jain–Koehler–Pham–Vuong: if $q$1 are exchangeable laws on $q$2 with $q$3 Lorentzian, then

$q$4

proved via concavity of $q$5 and the Donsker–Varadhan variational formula. This yields, for $q$6 with exchangeable Lorentzian law,

$q$7

which strictly refines the delete-one inequality by the nonnegative total correlation. The authors note this use of relative entropy contraction is new in the study of graph homomorphism inequalities, and remark that the contraction also follows from entropic independence via polarization.

Proof architecture

The argument proceeds by induction on $q$8 following the localization strategy of the authors' earlier work. After reducing to entrywise-positive $q$9 via a Perron–Frobenius perturbation $H$0 (which preserves antiferromagnetism because $H$1 is the unique positive eigenvalue), one conditions on the spin at a maximum-degree vertex $H$2 and applies the induction hypothesis, reducing the step to a local inequality involving $H$3 and its $H$4 neighbors. Via the dual set $H$5 of tangent hyperplanes to $H$6, whose log-convexity follows from a Cauchy–Schwarz argument, the local inequality reduces to a membership problem for each neighbor separately.

The main technical contribution resolves this membership problem for general antiferromagnetic models, replacing the model-specific low-dimensional calculations previously available only for $H$7 and $H$8. The quantity $H$9, where $I$0 is the supremum of $I$1, is realized as an optimized pressure $I$2 of an abstract spin system with pair-interaction energy $I$3. Two abstract properties — Gibbs laws of this system being exchangeable Lorentzian, and a deletion identity $I$4 — combine with the entropy inequality to give the monotonicity $I$5. Since the tangent-plane argument gives $I$6, monotonicity propagates $I$7 up to clique size $I$8, closing the induction.

Limitations and open questions

The paper is candid about the boundaries of its result. The biclique-maximizing conjecture for antiferromagnetic models remains open, so the upper half of the extremal picture is unproven. The class of clique-minimizing models strictly contains the antiferromagnetic class, as the tensor-product and zero-clique constructions show; a full characterization of clique-minimizing models is posed as an open problem. Equality in the vertex-inhomogeneous inequality is attained by disjoint unions of cliques, but uniqueness of extremizers under nondegeneracy assumptions, and stability of near-extremizers, are left unaddressed. Finally, the authors' conjecture that $I$9 is Lorentzian for every antiferromagnetic $G$0 only when $G$1 is a clique remains open, so the Lorentzian property established here may or may not characterize cliques among source graphs.

Conclusion

The paper establishes that the spectral condition of at most one positive eigenvalue — antiferromagnetism — is precisely what forces complete graphs to minimize homomorphism counts, in a vertex-inhomogeneous form that is sharp and, among antiferromagnetic models, equivalent to the spectral condition itself. Methodologically, it introduces relative entropy contraction for exchangeable Lorentzian laws and an abstract monotonicity principle for optimized pressure, converting Lorentzian log-concavity into extremal inequalities for partition functions. The result settles several standing conjectures and provides the lower bound of the anticipated clique/biclique extremal dichotomy, while leaving the biclique-maximizing conjecture and the characterization of clique-minimizing models open.