Eigenpolytope Universality and Graphical Designs

TL;DR

This paper demonstrates that eigenpolytopes of graphs can represent any polytope through affine transformations and extends graphical design theory to positively weighted graphs, establishing complexity results for minimal designs.

Contribution

It proves the universality of eigenpolytopes for all polytopes and links graphical designs to polytope faces via Gale duality, introducing new complexity results.

Findings

Eigenpolytopes are universal for all polytopes.

Graphical designs correspond to faces of eigenpolytopes.

Determining minimal graphical designs is computationally hard.

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.