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.