Counting faces of graphical zonotopes

TL;DR

This paper links the face enumeration of graphical zonotopes to graph colorings by expressing the $f$-polynomial as a specialized $q$-analog of the chromatic symmetric function, revealing new combinatorial connections.

Contribution

It establishes a novel relationship between the $f$-polynomial of graphical zonotopes and the chromatic symmetric function's $q$-analog, advancing understanding of zonotope face structures.

Findings

Number of vertices equals acyclic orientations.

$f$-polynomial is a principal specialization of a $q$-analog.

Connects zonotope face enumeration with graph coloring functions.

Abstract

It is a classical fact that the number of vertices of the graphical zonotope $Z_\Gamma$ is equal to the number of acyclic orientations of a graph $\Gamma$. We show that the $f$-polynomial of $Z_\Gamma$ is obtained as the principal specialization of the $q$-analog of the chromatic symmetric function of $\Gamma$.