Graphical designs and gale duality

A graphical design is a subset of graph vertices such that the weighted averages of certain graph eigenvectors over the design agree with their global averages. We use Gale duality to show that positively weighted graphical designs in regular graphs are in bijection with the faces of a generalized e...

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Bibliographic Details
Published inMathematical programming Vol. 200; no. 2; pp. 703 - 737
Main Authors Babecki, Catherine, Thomas, Rekha R.
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2023
Springer
Springer Nature B.V
Subjects
Apexes
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Eigenvectors
Full Length Paper
Graph theory
Graphs
Hypercubes
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Theoretical
Polytopes
Graph Sampling
Gale Duality
05C50
Stable Sets
52B35
Eigenpolytopes
Quadrature Rules
Graphical Designs
Graph Laplacian
Hamming Code
90C57
68R10
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Summary:A graphical design is a subset of graph vertices such that the weighted averages of certain graph eigenvectors over the design agree with their global averages. We use Gale duality to show that positively weighted graphical designs in regular graphs are in bijection with the faces of a generalized eigenpolytope of the graph. This connection can be used to organize, compute and optimize designs. We illustrate the power of this tool on three families of Cayley graphs – cocktail party graphs, cycles, and graphs of hypercubes – by computing or bounding the smallest designs that average all but the last eigenspace in frequency order.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-022-01861-0