SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 947-964, March 2024.
Abstract. 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.

中文翻译：

---

本征多面体普遍性和图形设计

SIAM 离散数学杂志，第 38 卷，第 1 期，第 947-964 页，2024 年 3 月。
摘要。我们证明，图的本征多面体是通用的，即每个多面体，直到仿射等价，都表现为某个正加权图的本征多面体。接下来，我们将图形设计理论（即图的正交规则）扩展到正加权图。通过多面体的盖尔对偶性，我们展示了图形设计和特征多面体的面之间的双射。这种双射证明了具有正正交权重和最小图形设计大小上限的图形设计的存在。将这种双射与特征多面体的普遍性联系起来，我们建立了三个复杂性结果：确定是否存在小于上述上限的图形设计是强 NP 完全的，找到最小图形设计是 NP 困难的，并且是 #P-complete 来计算最小图形设计的数量。