Simple polytopes arising from finite graphs

TL;DR

This paper classifies finite graphs based on the simplicity of their associated edge polytopes, linking geometric properties to algebraic structures and computing key invariants like Ehrhart polynomials.

Contribution

It provides a classification of graphs with simple edge polytopes and establishes a connection between simplicity and the existence of quadratic Gr"obner bases for their toric ideals.

Findings

Graphs with simple edge polytopes have toric ideals with quadratic Gr"obner bases.

Simple but not simplex edge polytopes are characterized as smooth but not simplices.

The Ehrhart polynomial and normalized volume of simple edge polytopes are explicitly computed.

Abstract

Let $G$ be a finite graph allowing loops, having no multiple edge and no isolated vertex. We associate $G$ with the edge polytope ${\cal P}_G$ and the toric ideal $I_G$. By classifying graphs whose edge polytope is simple, it is proved that the toric ideals $I_G$ of $G$ possesses a quadratic Gr\"obner basis if the edge polytope ${\cal P}_G$ of $G$ is simple. It is also shown that, for a finite graph $G$, the edge polytope is simple but not a simplex if and only if it is smooth but not a simplex. Moreover, the Ehrhart polynomial and the normalized volume of simple edge polytopes are computed.