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arXiv · 0804.4287

Simple polytopes arising from finite graphs

Hidefumi Ohsugi·Takayuki Hibi
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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.

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Hidefumi Ohsugi, Takayuki Hibi. 2008-04-27. Simple polytopes arising from finite graphs. https://arxiv.org/abs/0804.4287

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