Reflexive polytopes arising from bipartite graphs with γ-positivity associated to interior polynomials

In this paper, we introduce polytopes BG arising from root systems Bn and finite graphs G, and study their combinatorial and algebraic properties. In particular, it is shown that BG is reflexive if and only if G is bipartite. Moreover, in the case, BG has a regular unimodular triangulation. This implies that the h-polynomial of BG 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 BG 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.