Reflexive polytopes arising from bipartite graphs with $$\gamma $$-positivity associated to interior polynomials

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