Ehrhart series of fractional stable set polytopes of finite graphs

Abstract

The fractional stable set polytope $\mathrm{FRAC}(G)$ of a simple graph $G$ with $d$ vertices is a rational polytope that is the set of nonnegative vectors $(x_1,\dots ,x_d)$ 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 $2\mathrm{FRAC}(G)$ is alternatingly increasing; (ii) The Ehrhart ring of $\mathrm{FRAC}(G)$ is Gorenstein; (iii) The coefficients of the numerator of the Ehrhart series of $\mathrm{FRAC}(G)$ are symmetric, unimodal and computed by the $\delta$-vector of $2\mathrm{FRAC}(G)$.

12 pages, V2: some minor revisions, references added