Uniformly Weighted Graphical Designs
Discrete Mathematics
Summary
The authors study a special way to pick groups of points (vertices) in a graph along with equal weights so that they nicely average certain functions on the graph, called uniformly weighted graphical designs. They explain when such equal-weight designs can or cannot exist and give examples of graphs that have them and those that don't. Their work also connects these designs to well-known math objects and uses geometry to prove some important properties. Additionally, they create new examples of graphs where these equal-weight designs do not exist by examining the graph's Laplacian polynomial.
Authors
Zawad Chowdhury, Rekha R. Thomas
Abstract
A graphical design is a subset of vertices of a graph, along with a weight for each chosen vertex, that can perfectly average chosen subspaces of functions on the graph. A design is uniformly weighted if all the weights are equal, and several well-known combinatorial objects such as orthogonal arrays, combinatorial block designs and t-wise permutations are uniformly weighted graphical designs. While one might expect to see uniformly weighted designs in structured graphs, they do not always exist. In this paper we characterize the existence of uniformly weighted graphical designs, and use our result to provide several families of graphs that have, and do not have, such designs. Our results offer a polyhedral view of the structures that control the existence and cardinalities of these designs. In particular, we characterize all uniformly weighted designs of threshold graphs, and provide a geometric proof of the duality of linear codes and linear orthogonal arrays. We also provide a novel construction for graphs whose Laplacian characteristic polynomials are almost irreducible, to produce families without uniformly weighted designs.