The Polyhedral Geometry of Graphical Designs

Abstract

A graphical design is a quadrature rule for a graph. That is, a graphical design is a subset of graph vertices for which the global averages of certain Laplacian eigenvectors are equal to weighted averages of these vectors over the design subset. This definition was inspired by classical quadrature rules on the sphere. This thesis refines and extends the initial definition of graphical designs, which was left with ambiguity, to graphs with positive edge weights. Through Gale duality, we establish a bijection between graphical designs and the face lattices of the eigenpolytopes of a graph. This connection proves the existence of positively weighted graphical designs averaging any collection of eigenvectors of a graph, and provides methods to compute, organize, and optimize graphical designs. These tools are then applied to three families of graphs: cocktail party graphs, cycles, and cubes. This thesis also considers complexity and algorithms for related computational questions. We show the universality of eigenpolytopes for positively weighted graphs; that is, every polytope up to affine equivalence appears as the eigenpolytope of a positively weighted graph, and we provide a strongly polynomial time algorithm for this construction. As a stepping stone, we show that given an appropriate orthogonal basis of $\mathbb{R}^n$ and a partition of these basis vectors, there is a positively weighted graph which has Laplacian eigenspaces specified by this basis and partition. These algorithms establish the first complexity results for graphical designs, piggybacking off of complexity results for polytopes.