Ehrhart series of fractional stable set polytopes of finite graphs

TL;DR

This paper investigates the Ehrhart series of fractional stable set polytopes of graphs, revealing their algebraic and combinatorial properties such as symmetry, unimodality, and Gorenstein conditions.

Contribution

It establishes new properties of the Ehrhart series and $ ext{delta}$-vector of fractional stable set polytopes, including Gorensteinness and alternatingly increasing $ ext{delta}$-vector.

Findings

The $ ext{delta}$-vector of $2 ext{FRAC}(G)$ is alternatingly increasing.

The Ehrhart ring of $ ext{FRAC}(G)$ is Gorenstein.

The numerator coefficients of the Ehrhart series are symmetric and unimodal.

Abstract

The fractional stable set polytope ${\rm FRAC}(G)$ of a simple graph $G$ with $d$ vertices is a rational polytope that is the set of nonnegative vectors $(x_1,\ldots,x_d)$ satisfying $x_i+x_j\le 1$ for every edge $(i,j)$ of $G$. In this paper we show that (i) The $\delta$-vector of a lattice polytope $2 {\rm FRAC}(G)$ is alternatingly increasing; (ii) The Ehrhart ring of ${\rm FRAC}(G)$ is Gorenstein; (iii) The coefficients of the numerator of the Ehrhart series of ${\rm FRAC}(G)$ are symmetric, unimodal and computed by the $\delta$-vector of $2 {\rm FRAC}(G)$.