In this paper, we introduce polytopes \({\mathscr {B}}_G\) arising from root systems \(B_n\) and finite graphs G, and study their combinatorial and algebraic properties. In particular, it is shown that \({\mathscr {B}}_G\) is reflexive if and only if G is bipartite. Moreover, in the case, \({\mathscr {B}}_G\) has a regular unimodular triangulation. This implies that the \(h^*\)-polynomial of \({\mathscr {B}}_G\) is palindromic and unimodal when G is bipartite. Furthermore, we discuss stronger properties, namely the \(\gamma \)-positivity and the real-rootedness of the \(h^*\)-polynomials. In fact, if G is bipartite, then the \(h^*\)-polynomial of \({\mathscr {B}}_G\) is \(\gamma \)-positive and its \(\gamma \)-polynomial is given by an interior polynomial (a version of the Tutte polynomial for a hypergraph). The \(h^*\)-polynomial is real-rooted if and only if the corresponding interior polynomial is real-rooted. From a counterexample to Neggers–Stanley conjecture, we construct a bipartite graph G whose \(h^*\)-polynomial is not real-rooted but \(\gamma \)-positive, and coincides with the h-polynomial of a flag triangulation of a sphere.

中文翻译：

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由与内部多项式相关的$$ \γ$$γ正性的二部图产生的自反多面体

在本文中，我们介绍了由根系统\（B_n \）和有限图G引起的多拓扑\（{\ mathscr {B}} _ G \），并研究了它们的组合和代数性质。特别地，证明了\（{\ mathscr {B}} _ G \）是自反的，并且仅当G是二分的时才如此。此外，在这种情况下，\（{\ mathscr {B}} _ G \）具有规则的单模三角剖分。这意味着，当G为二分式时，\（{\ mathscr {B}} _ G \）的\（h ^ * \）-多项式是回文的且是单峰的。此外，我们讨论了更强的特性，即\（\ gamma \）的正性和实根\（h ^ * \）-多项式 实际上，如果G为二分式，则\（{\ mathscr {B}} _ G \）的\（h ^ * \）-多项式为\（\ gamma \）为正，而其\（\ gamma \） -多项式由内部多项式（超图的Tutte多项式的一种形式）给出。仅当对应的内部多项式是实数根时，\（h ^ * \） -多项式才是实数根。从Neggers–Stanley猜想的反例中，我们构造一个二部图G，其\（h ^ * \）-多项式不是实根的，而是\（\ gamma \）为正，并且与h重合。-球体的三角剖分的多项式。