Roots of Ehrhart polynomials arising from graphs

Abstract

Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck {\it et al.}\ that all roots $\alpha$ of Ehrhart polynomials of polytopes of dimension $D$ satisfy $-D\le \Re (\alpha )\le D-1$, but also reveals some interesting phenomena for each type of polytope. Here we present two new conjectures: (1) the roots of the Ehrhart polynomial of an edge polytope for a complete multipartite graph of order $d$ lie in the circle $|z+\frac{d}{4}|\le \frac{d}{4}$ or are negative integers, and (2) a Gorenstein Fano polytope of dimension $D$ has the roots of its Ehrhart polynomial in the narrower strip $-\frac{D}{2}\le \Re (\alpha )\le \frac{D}{2}-1$. Some rigorous results to support them are obtained as well as for the original conjecture. The root distribution of Ehrhart polynomials of each type of polytope is plotted in figures.

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• The $h^{\ast }$-polynomials of locally anti-blocking lattice polytopes and their $\gamma$-positivity
• On the gamma-vector of symmetric edge polytopes
• Reflexive polytopes arising from bipartite graphs with $\gamma$-positivity associated to interior polynomials
• Directed acyclic decomposition of Kuramoto equations
• Facets of Random Symmetric Edge Polytopes, Degree Sequences, and Clustering
• Gorenstein matroids
• Ehrhart polynomial roots of reflexive polytopes
• $h^{\ast }$-vectors of graph polytopes using activities of dissecting spanning trees
• Normalized Volumes of Type-PQ Adjacency Polytopes for Certain Classes of Graphs
• Rigid Gorenstein toric Fano varieties arising from directed graphs