We extend the theory of graphical designs, which are quadrature rules for graphs, to positively weighted graphs. Through Gale duality for polytopes, we show that there is a bijection between graphical designs and the faces of eigenpolytopes associated to the graph. This bijection proves the existence of graphical designs with positive quadrature weights, and upper bounds the size of a graphical design. We further show that any combinatorial polytope appears as the eigenpolytope of a positively weighted graph. Through this universality, we establish two complexity results for graphical designs: it is strongly NP-complete to determine if there is a graphical design smaller than the mentioned upper bound, and it is #P-complete to count the number of minimal graphical designs.

翻译：我们将图形设计理论( 图形的二次曲线规则) 扩展到正加权的图形。 我们通过多面图的 Gale 双轨制, 显示图形设计与图形相联的egenpolytops 脸部之间有两条线。 这个双轨制证明了存在带有正二次曲线重量的图形设计, 以及图形设计大小的上界。 我们进一步显示, 任何组合式的多轨制都显示为正加权图形的二次曲线。 通过这一普遍性, 我们为图形设计确定了两个复杂的结果: 确定是否有比上述的上界小的图形设计, 并且计算最小的图形设计数量是 # P- 完整的 。