Let $G$ be a finite graph on the vertex set $[d] = \{1, ..., d \}$ with the
edges $e_1, ..., e_n$ and $K[\tb] = K[t_1, ..., t_d]$ the polynomial ring in
$d$ variables over a field $K$. The edge ring of $G$ is the semigroup ring
$K[G]$ which is generated by those monomials $\tb^e = t_it_j$ such that $e =
\{i, j\}$ is an edge of $G$. Let $K[\xb] = K[x_1, ..., x_n]$ be the polynomial
ring in $n$ variables over $K$ and define the surjective homomorphism $\pi :
K[\xb] \to K[G]$ by setting $\pi(x_i) = \tb^{e_i}$ for $i = 1, ..., n$. The
toric ideal $I_G$ of $G$ is the kernel of $\pi$.

Given arbitrary integers $k$ and $d$ with $0 \leq 2k \leq d$, we construct a
Gorenstein Fano polytope $\Pc \subset \RR^d$ of dimension $d$ such that (i) its
Ehrhart polynomial $i(\Pc, n)$ possesses $d$ distinct roots; (ii) $i(\Pc, n)$
possesses exactly $2k$ imaginary roots; (iii) $i(\Pc, n)$ possesses exactly $d
- 2k$ real roots; (iv) the real part of each of the imaginary roots is equal to
$- 1 / 2$; (v) all of the real roots belong to the open interval $(-1, 0)$.

A $\delta$-vector $\delta(\Pc)= (\delta_0, \delta_1, ..., \delta_d)$ is
called shifted symmetric if $\delta_{d-i} = \delta_{i+1}$ for each $0 \leq i
\leq [(d-1)/2]$. A natural family of $(0,1)$-polytopes with shifted symmetric
$\delta$-vectors will be studied.