Ehrhart series for Connected Simple Graphs

TL;DR

This paper investigates the Ehrhart ring of edge polytopes for connected simple graphs, providing a combinatorial description of the defining ideal for graphs not satisfying the odd cycle condition, and explores factoring properties of the Ehrhart series.

Contribution

It offers a combinatorial generating set for the Ehrhart ring's ideal for graphs without the odd cycle condition and derives new factoring properties of the Ehrhart series.

Findings

Provides a combinatorial description of the defining ideal for non-odd cycle graphs.

Establishes two new factoring properties of the Ehrhart series.

Determines the root distribution of Ehrhart polynomials for bipartite polygon trees.

Abstract

The Ehrhart ring of the edge polytope $\mathcal{P}_G$ for a connected simple graph $G$ is known to coincide with the edge ring of the same graph if $G$ satisfies the odd cycle condition. This paper gives for a graph which does not satisfy the condition, a generating set of the defining ideal of the Ehrhart ring of the edge polytope, described by combinatorial information of the graph. From this result, two factoring properties of the Ehrhart series are obtained; the first one factors out bipartite biconnected components, and the second one factors out a even cycle which shares only one edge with other part of the graph. As an application of the factoring properties, the root distribution of Ehrhart polynomials for bipartite polygon trees is determined.