The Inverse Symplectic Eigenvalue Problem of a Graph

TL;DR

This work defines and analyzes the inverse symplectic eigenvalue problem for labeled graphs, building a toolkit centered on the Strong Symplectic Spectral Property (SSSP) and its consequences. By introducing the Supergraph Theorem, Bifurcation Theorem, Matrix Liberation Lemma, and coupled zero forcing, the authors solve the ISEP-$G$ for all graphs of order four and establish broad families of symplectic spectrally arbitrary graphs, including both sparse and dense structures. They characterize sympPD matrices, derive sparsity bounds, and construct explicit graph families (e.g., triangular paths, coronas, and couplings) that realize any prescribed symplectic spectrum under suitable labeling. The results have implications for understanding how graph structure governs symplectic spectra, enabling systematic design of matrices with desired symplectic eigenvalues in applications related to Hamiltonian dynamics and quantum information.

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.

Figures (5)

• Figure 1.1: Two labelings $L$ and $\sigma(L)$ of $P_4$ where the matrices $N\in \mathcal{S}_+(P_4^L)$ and $P_\sigma NP_{\sigma}^{\top}\in \mathcal{S}_+(P_4^{\sigma(L)})$ have different symplectic eigenvalues.
• Figure 2.1: Standard labeled triangular path $\operatorname{TP}_{2p}^I$ when $p$ is odd.
• Figure 2.2: Standard labeled triangular path $\operatorname{TP}_{2p}^I$ when $p$ is even.
• Figure 5.1: For $i=1,2,3$, a representative labeling $L_i$ associated with the coupling $\mathfrak{C}_i$, as defined in Remark \ref{['r:ord4couplings']}.
• Figure 6.1: A caterpillar with a perfect matching (black edges form a perfect matching).

Theorems & Definitions (101)

• Theorem 1.1: Williamson's Theorem
• Proposition 1.2
• proof
• Example 1.3
• Example 1.4
• Remark 2.1
• Example 2.2
• Example 2.3
• Theorem 2.4
• Proposition 2.5