The Inverse Symplectic Eigenvalue Problem of a Graph

TL;DR

This paper investigates the inverse symplectic eigenvalue problem for positive definite matrices associated with graphs, introducing new theoretical tools and solving it for specific graph families and all graphs of order four.

Contribution

It develops novel tools like the SSSP, Supergraph Theorem, and Matrix Liberation Lemma, and solves the inverse symplectic eigenvalue problem for various graph classes.

Findings

Solved the inverse symplectic eigenvalue problem for several graph families.

Established a sharp lower bound on the number of nonzero entries in symplectic positive definite matrices.

Introduced new tools such as the SSSP and graph coupling methods.

Abstract

Symplectic geometry plays an increasingly important role in mathematics, physics and applications, and naturally gives rise to interesting matrix families and properties. One of these is the notion of symplectic eigenvalues, whose existence for positive definite matrices is known as Williamson's theorem or decomposition. This notion of symplectic eigenvalues gives rise to inverse problems. We introduce the inverse symplectic eigenvalue problem for positive definite matrices described by a labeled graph and solve it for several families of labeled graphs and all labeled graphs of order four. To solve these problems we develop various tools such as the Strong Symplectic Spectral Property (SSSP) and its consequences such as the Supergraph Theorem, the Bifurcation Theorem, and the Matrix Liberation Lemma for symplectic eigenvalues, graph couplings to describe collections of labelings of a graph that produce the same symplectic eigenvalues, and coupled graph zero forcing. We establish numerous results for symplectic positive definite matrices, including a sharp lower bound on the number of nonzero entries of such a matrix (or equivalently, the number of edges in its graph). This lower bound is a consequence of a lower bound on the sum of number of nonzero entries in an irreducible positive definite matrix and its inverse.