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.

翻译：我们证明，在仿射等价的意义下，每个多胞体都可以作为某个正权图的特征多胞体出现，因此图特征多胞体具有普适性。接着，我们将图设计理论（即图的求积规则）推广到正权图上。通过多胞体的Gale对偶性，我们揭示了图设计与特征多胞体的面之间的双射关系。这一双射证实了具有正求积权重的图设计的存在性，并给出了最小图设计规模的上界。将该双射与特征多胞体的普适性相关联，我们建立了三个复杂性结果：判断是否存在比上述上界更小的图设计是强NP完全问题；寻找最小图设计是NP难问题；计数最小图设计的数量是#P完全问题。