# On reconstruction of eigenfunctions of Johnson graphs

## TL;DR

This paper investigates conditions under which eigenfunctions of Johnson graphs can be uniquely reconstructed from their values on a sphere, providing necessary and sufficient criteria and illustrating cases of failure.

## Contribution

It establishes exact numerical conditions for the unique reconstruction of eigenfunctions of Johnson graphs from sphere data, advancing understanding of their structure.

## Key findings

- Necessary and sufficient conditions for unique reconstruction
- Examples of non-uniqueness when conditions fail
- Analysis applicable for large n in Johnson graphs

## Abstract

In the present work we consider the problem of a reconstruction of eigenfunctions of the Johnson graph $J(n,w)$. We give necessary and sufficient numerical conditions for a unique reconstruction of an eigenfunction with given eigenvalue by its values on a sphere of given radius $r$ for $n$ big enough. We also provide examples of functions equal on the sphere but not equal on the full vertex set in the case of a failure of these conditions.