Symmetric edge polytopes and matching generating polynomials

Abstract

Symmetric edge polytopes ${\mathcal{A}}_G$ of type A are lattice polytopes arising from the root system $A_n$ and finite simple graphs $G$. There is a connection between ${\mathcal{A}}_G$ and the Kuramoto synchronization model in physics. In particular, the normalized volume of ${\mathcal{A}}_G$ plays a central role. In the present paper, we focus on a particular class of graphs. In fact, for any cactus graph $G$, we give a formula for the $h^{\ast }$-polynomial of ${\mathcal{A}}_{\hat{G}}$ by using matching generating polynomials, where $\hat{G}$ is the suspension of $G$. This gives also a formula for the normalized volume of ${\mathcal{A}}_{\hat{G}}$. Moreover, via the chemical graph theory, we show that for any cactus graph $G$, the $h^{\ast }$-polynomial of ${\mathcal{A}}_{\hat{G}}$ is real-rooted. Finally, we extend the discussion to symmetric edge polytopes of type $B$, which are lattice polytopes arising from the root system $B_n$ and finite simple graphs.

arXiv comment18 pages, to appear in Combinatorial Theory

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• On the gamma-vector of symmetric edge polytopes
• PQ-type adjacency polytopes of join graphs
• Facets of Random Symmetric Edge Polytopes, Degree Sequences, and Clustering
• Normalized Volumes of Type-PQ Adjacency Polytopes for Certain Classes of Graphs
• Rigid Gorenstein toric Fano varieties arising from directed graphs