Facets and facet subgraphs of symmetric edge polytopes

TL;DR

This paper explores the relationship between the geometric facets of symmetric edge polytopes and the topological structure of the underlying graphs, revealing a complete correspondence with maximal bipartite subgraphs.

Contribution

It provides a complete characterization of the correspondence between facets of symmetric edge polytopes and maximal bipartite subgraphs of the underlying graph.

Findings

Facets of symmetric edge polytopes correspond to maximal bipartite subgraphs.

The study bridges geometric and combinatorial graph properties.

Results apply to algebraic and geometric contexts involving these polytopes.

Abstract

Symmetric edge polytopes, a.k.a. PV-type adjacency polytopes, associated with undirected graphs have been defined and studied in several seemingly independent areas including number theory, discrete geometry, and dynamical systems. In particular, the authors are motivated by the study of the algebraic Kuramoto equations of unmixed form whose Newton polytopes are the symmetric edge polytopes. The interplay between the geometric structure of symmetric edge polytopes and the topological structure of the underlying graphs has been a recurring theme in recent studies. In particular, ``facet/face subgraphs'' have emerged as one of the central concepts in describing this symmetry. Continuing along this line of inquiry we provide a complete description of the correspondence between facets/faces of a symmetric edge polytope and maximal bipartite subgraphs of the underlying connected graph.