Graphical Designs find Combinatorial Structures

TL;DR

This paper explores the relationship between graphical designs and well-known combinatorial structures in highly-structured graphs, revealing new connections and applications of spectral graph theory.

Contribution

It demonstrates that graphical designs can correspond to classical combinatorial objects in specific graphs, bridging spectral graph theory and combinatorics.

Findings

Graphical designs in hypercube graphs relate to orthogonal arrays.

In Johnson graphs, they connect to block designs and extremizers of the Erdos-Ko-Rado theorem.

Certain designs in Mycielskian graphs align with original graph designs.

Abstract

Graphical designs are subsets of vertices of a graph that perfectly average a selected set of eigenvectors of the Graph Laplacian. We show that in highly-structured graphs, graphical designs can coincide with highly structured and well-known combinatorial objects: orthogonal arrays in hypercube graphs, combinatorial block designs and extremizers of the Erdos-Ko-Rado theorem in Johnson graphs, and t-wise uniform sets of permutations and symmetric subgroups in normal Cayley graphs on the symmetric group. These connections allow tools from spectral graph theory to bear on these combinatorial objects. We also show that the central vertex in a Mycielskian is an extremely good design and certain designs of the Mycielskian coincide with designs of the original graph.