Symmetric edge polytopes and matching generating polynomials

TL;DR

This paper studies symmetric edge polytopes associated with cactus graphs, providing formulas for their $h^*$-polynomials and normalized volumes, and proves real-rootedness of these polynomials, extending results to type B polytopes.

Contribution

It introduces explicit formulas for $h^*$-polynomials of symmetric edge polytopes of cactus graphs using matching generating polynomials and establishes their real-rootedness, also extending to type B polytopes.

Findings

Formulas for $h^*$-polynomials of $ ext{A}_G$ for cactus graphs

Normalized volume formulas derived from $h^*$-polynomials

Proof of real-rootedness of the $h^*$-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 $h^*$-polynomial of $\mathcal{A}_{\widehat{G}}$ by using matching generating polynomials, where $\widehat{G}$ is the suspension of $G$. This gives also a formula for the normalized volume of $\mathcal{A}_{\widehat{G}}$. Moreover, via the chemical graph theory, we show that for any cactus graph $G$, the $h^*$-polynomial of $\mathcal{A}_{\widehat{G}}$ 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.