Structural Classes for Chollet's Permanent Conjecture

Abstract

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 (entrywise) product. In this paper, we study natural structural classes for which the conjecture holds. We first show that a stronger inequality holds for a broad class of matrices with bipartite support. We then introduce a simple way of joining positive semidefinite matrices and give conditions under which Chollet's inequality is preserved under this operation. For graph Laplacians, this operation corresponds to vertex coalescence and gives larger structured graph families satisfying Chollet's inequality from simpler graph classes.

arXiv commentMajor revision. The results are now developed in a more general matrix setting, with stronger statements, new structural classes, and expanded proofs