  In this paper, we introduce polytopes ${\mathcal B}_G$ arising from root
systems $B_n$ and finite graphs $G$, and study their combinatorial and
algebraic properties. In particular, it is shown that ${\mathcal B}_G$ is
reflexive if and only if $G$ is bipartite. Moreover, in the case, ${\mathcal
B}_G$ has a regular unimodular triangulation. This implies that the
$h^*$-polynomial of ${\mathcal B}_G$ 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 ${\mathcal B}_G$ 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.
