论文标题
对称边缘多型和匹配生成多项式
Symmetric edge polytopes and matching generating polynomials
论文作者
论文摘要
对称的边缘polytopes $ \ MATHCAL {A} _g $ A类型是源自根系$ a_n $和有限的简单图$ g $的晶格polytopes。 $ \ Mathcal {a} _g $与物理学中的Kuramoto同步模型之间存在连接。特别是，$ \ Mathcal {a} _g $的归一化体积起着核心作用。在本文中，我们专注于特定的图表。实际上，对于任何仙人掌图$ g $，我们为$ h^*$ - $ \ MATHCAL {a} _ {\ wideHat {g}} $的$ h^*$ - 多项式提供了使用匹配的生成多项式，其中$ \ wideHat {g} $是$ g $的悬挂。这还提供了$ \ mathcal {a} _ {\ wideHat {g}} $的归一量化卷的公式。此外，通过化学图理论，我们表明，对于任何仙人掌图$ g $，$ h^*$ - $ \ MATHCAL的多项式{a} _ {\ wideHat {g}} $是实地的。最后，我们将讨论扩展到类型$ b $的对称边缘多面体，它们是源自根系$ b_n $和有限简单图的晶格多型。
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.
