Between graphical zonotope and graph-associahedron

TL;DR

This paper introduces a family of polytopes interpolating between the graphical zonotope and the graph-associahedron, linking combinatorial graph structures with geometric objects through Hopf algebra morphisms.

Contribution

It defines a new collection of generalized permutohedra associated to graphs, connecting graph theory, polytope theory, and algebraic combinatorics in a novel way.

Findings

Defined a finite collection of generalized permutohedra for simple graphs.

Connected weighted integer point enumeration to Hopf algebra morphisms.

Established a geometric-combinatorial-algebraic framework linking graphs and polytopes.

Abstract

This manuscript introduces a finite collection of generalized permutohedra associated to a simple graph. The first polytope of this collection is the graphical zonotope of the graph and the last is the graph-associahedron associated to it. We describe the weighted integer points enumerators for polytopes in this collection as Hopf algebra morphisms of combinatorial Hopf algebras of decorated graphs.