Graphical zonotopes with the same face vector

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.