Minimum supports of eigenfunctions of Johnson graphs

TL;DR

This paper investigates the minimum number of nonzero entries in eigenvectors of Johnson graphs, establishing lower bounds and characterizing eigenvectors that achieve these bounds for large n.

Contribution

It provides new lower bounds on the support size of eigenvectors of Johnson graphs and characterizes those that attain these bounds for sufficiently large n.

Findings

Eigenvectors with eigenvalue λ_i have at least 2^i times a binomial coefficient nonzeros.

Characterization of eigenvectors that meet the minimum support bound.

Results hold for sufficiently large n, depending on i and w.

Abstract

We study the weights of eigenvectors of the Johnson graphs $J(n,w)$. For any $i \in \{1,\ldots,w\}$ and sufficiently large $n, n\geq n(i,w)$ we show that an eigenvector of $J(n,w)$ with the eigenvalue $\lambda_i=(n-w-i)(w-i)-i$ has at least $2^i(^{n-2i}_{w-i})$ nonzeros and obtain a characterization of eigenvectors that attain the bound.