  We show that the eigenpolytopes of graphs are universal in the sense that
every polytope, up to affine equivalence, appears as the eigenpolytope of some
positively weighted graph. We next extend the theory of graphical designs,
which are quadrature rules for graphs, to positively weighted graphs. Through
Gale duality for polytopes, we show a bijection between graphical designs and
the faces of eigenpolytopes. This bijection proves the existence of graphical
designs with positive quadrature weights, and upper bounds the size of a
minimal graphical design. Connecting this bijection with the universality of
eigenpolytopes, we establish three complexity results: it is strongly
NP-complete to determine if there is a graphical design smaller than the
mentioned upper bound, it is NP-hard to find a smallest graphical design, and
it is #P-complete to count the number of minimal graphical designs.
