On Chollet's Permanent Conjecture for Graph Laplacians

In 1982, Chollet conjectured that $\mathrm{p}\mathrm{e}\mathrm{r}(A\circ B)\le \mathrm{p}\mathrm{e}\mathrm{r}(A)\mathrm{p}\mathrm{e}\mathrm{r}(B)$ 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 $\mathrm{p}\mathrm{e}\mathrm{r}(A\circ A)\le \mathrm{p}\mathrm{e}\mathrm{r}(A{)}^2$. We prove $\mathrm{p}\mathrm{e}\mathrm{r}(A\circ A)\le \mathrm{p}\mathrm{e}\mathrm{r}(A{)}^2$ for symmetric $Z$-matrices with nonnegative diagonal whose support graph is bipartite. Motivated by this, we study the Laplacian inequality $\mathrm{p}\mathrm{e}\mathrm{r}(L_G\circ L_G)\le \mathrm{p}\mathrm{e}\mathrm{r}(L_G{)}^2$ 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 $\mathrm{\#}P$-hard quantity.