Structure and Complexity of Graphical Designs for Weighted Graphs through Eigenpolytopes

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.