AI 中文总结
该研究证明反铁磁图满足图同态的下界不等式，其顶点非齐次形式统一推广了多个相关下界结果，同时证实了相关猜想，并通过强化Shearer不等式的删除形式为图同态不等式提供了新方法。
AI 中文摘要
若边赋权图H（允许有环）的邻接矩阵逐元素非负且正特征值个数（计重数）不超过1，则称H是反铁磁的。我们证明，对任意图G，设d_v为G中顶点v的度数，当H是反铁磁图时，有hom(G,H) ≥ ∏_{v∈V(G)} hom(K_{d_v+1},H)^(1/(d_v+1))。事实上，我们证明了该不等式的顶点非齐次强化形式，允许在G的每个顶点处使用不同的逸度向量。这一结果统一推广了Sah、Sawhney、Stoner和Zhao关于独立集的下界不等式，Csikvári关于q着色的下界不等式，以及本文作者关于至多两种真颜色的半真着色的下界不等式。此外，它证实了本文作者以及Davies和LeBlanc近期提出的猜想。一个具有独立意义的关键要素是Shearer不等式的删除一种形式对洛伦兹测度的强化，这为图同态不等式提供了新的研究途径。
英文摘要
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 $d_v:=\operatorname{deg}_G(v)$, $$\operatorname{hom}(G,H) \ge \prod_{v\in V(G)} \operatorname{hom}(K_{d_v+1},H)^{\frac{1}{d_v+1}},$$ 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.
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