Simple polytopes arising from finite graphs

Abstract

Let $G$ be a finite graph allowing loops, having no multiple edge and no isolated vertex. We associate $G$ with the edge polytope ${\mathcal{P}}_G$ 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 Gröbner basis if the edge polytope ${\mathcal{P}}_G$ 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.

11 pages

Cited by in corpus (2)

• The number of edges of the edge polytope of a finite simple graph
• Decomposable edge polytopes of finite graphs