Graphical zonotopes with the same face vector

TL;DR

This paper introduces a flip operation on graphs that allows constructing combinatorially distinct graphical zonotopes sharing identical face vectors, facilitating the study of their combinatorial properties.

Contribution

The paper presents a new quadrilateral flip operation on graphs and proves it preserves face vectors of the associated graphical zonotopes, enabling systematic construction of such zonotopes.

Findings

Flip operation preserves face vectors of graphical zonotopes.

All triangulations of an n-gon are connected via flips, sharing the same face vector.

Explicit computation of face vectors and total face counts for these zonotopes.

Abstract

We are interested in constructing zonotopes which are combinatorially nonequivalent but have the same face vector. In this paper we introduce a quadrilateral flip operation on graphs. We show that, if one graph is obtained from another graph by a flip, then the face vectors of the graphical zonotopes of these two graphs are the same. In this way, we can easily construct a class of combinatorially nonequivalent graphical zonotopes which share the same face vector. It is known that all triangulations of the n-gon are connected through the flip operation. Thus their graphical zonotopes have the same face vector. We will compute this vector and the total number of faces.