The Algebraic Boundary of Graph Elliptopes

TL;DR

This paper characterizes the algebraic boundary of graph elliptopes, especially for cycle completable graphs, revealing its structure as determinantal hypersurfaces and Lissajous varieties, and relates it to spectrahedral properties.

Contribution

It provides a complete algebraic boundary description for cycle completable graph elliptopes and introduces the cycle polynomial's role in this characterization, including an inductive computation method.

Findings

The algebraic boundary is a union of determinantal hypersurfaces and Lissajous varieties for cycle completable graphs.

The boundary is disjoint from the interior iff the elliptope is a spectrahedron, i.e., the graph is chordal.

The cycle polynomial's degree is determined, settling an open question.

Abstract

This paper studies the algebraic boundary of the elliptope $\mathcal{E}(G)$ of a graph $G$. In particular, we completely characterize the algebraic boundary of $\mathcal{E}(G)$ when $G$ is cycle completable. In this case, the boundary is a union of determinantal hypersurfaces and Lissajous varieties, i.e., images of rational linear subspaces under the coordinatewise cosine map. As an application, we show that the algebraic boundary of $\mathcal{E}(G)$ is disjoint from its interior precisely when $\mathcal{E}(G)$ is a spectrahedron or, equivalently, when $G$ is a chordal graph. A central ingredient for the defining equation of the boundary hypersurface is the cycle polynomial, which captures the algebraic boundary of the elliptope $\mathcal{E}(C_n)$ of the $n$-th cycle graph $C_n$. We show that the cycle polynomial of $C_n$ is the resultant of two smaller cycle polynomials. Via this result, Sylvester's determinantal formula offers an inductive method for computing the cycle polynomial which mirrors a geometric property of metric polytopes. We also determine the degree of the homogeneous cycle polynomial, settling an open question of Sturmfels and Uhler (2010).