Root systems and graph associahedra

TL;DR

This paper characterizes when the facet vectors of a graph associahedron form a root system, showing it occurs only for cycle graphs and results in a type A root system.

Contribution

It establishes a precise condition linking graph structure to root systems, specifically identifying cycle graphs as the unique case.

Findings

Facet vectors form a root system only for cycle graphs.

The resulting root system is of type A.

Provides a complete characterization of this property.

Abstract

It is known that a connected simple graph $G$ associates a simple polytope $P_G$ called a graph associahedron in Euclidean space. In this paper we show that the set of facet vectors of $P_G$ forms a root system if and only if $G$ is a cycle graph and that the root system is of type A.