Uniformly Weighted Graphical Designs
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.
Related
- A weighted casebase framework for outcome-dependent two-phase sampling designs.
- Learning extremal graphical structures in high dimensions
- A graphical approach to integration and the fundamental theorem
- Degree of the Gaussian graphical model of the cycle
- Interior curvature estimates for graphical curvature quotient equations
BibTeX
@misc{chowdhury2026uniformly,
title = {{Uniformly Weighted Graphical Designs}},
author = {Zawad Chowdhury and Rekha R. Thomas},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27624},
eprint = {2608.27624},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27624},
}
RIS
TY - UNPB
TI - Uniformly Weighted Graphical Designs
AU - Zawad Chowdhury
AU - Rekha R. Thomas
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27624
UR - https://arxiv.org/abs/2608.27624
AB - 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.
ER -