In 1982, Chollet conjectured that $\mathrm{per}(A\circ B)\le \mathrm{per}(A)\mathrm{per}(B)$ for Hermitian positive semidefinite matrices $A,B$, where $\circ$ denotes the Hadamard product, and observed that in the real symmetric case it suffices to prove $\mathrm{per}(A\circ A)\le \mathrm{per}(A)^2$. We prove $\mathrm{per}(A\circ A)\le \mathrm{per}(A)^2$ for symmetric $Z$-matrices with nonnegative diagonal whose support graph is bipartite. Motivated by this, we study the Laplacian inequality $\mathrm{per}(L_G\circ L_G)\le \mathrm{per}(L_G)^2$ for the graph Laplacian $L_G$. We introduce a compositional framework for permanental inequalities on graph Laplacians, showing that Chollet's inequality is preserved under vertex coalescence. This enables the extension of the inequality from basic graph classes to large structured families, revealing new tractable regimes for a fundamentally $\#P$-hard quantity.

翻译：1982年，Chollet猜想对于埃尔米特正半定矩阵 $A,B$ 有 $\mathrm{per}(A\circ B)\le \mathrm{per}(A)\mathrm{per}(B)$，其中 $\circ$ 表示Hadamard乘积，并指出在实对称情形下只需证明 $\mathrm{per}(A\circ A)\le \mathrm{per}(A)^2$ 即可。本文证明了对于对角元非负且支撑图为二分图的对称 $Z$-矩阵，有 $\mathrm{per}(A\circ A)\le \mathrm{per}(A)^2$ 成立。受此启发，我们研究了图拉普拉斯算子 $L_G$ 的拉普拉斯不等式 $\mathrm{per}(L_G\circ L_G)\le \mathrm{per}(L_G)^2$。我们引入了一种关于图拉普拉斯算子永久性不等式的组合框架，证明了Chollet不等式在顶点并合下保持不变。这使得该不等式能够从基本图类扩展到大型结构化图族，揭示了针对本质上是 $\#P$-难问题的新可解区域。