Eigenvalues and eigenfunctions of a Hamming ball

TL;DR

This paper characterizes the eigenvalues and eigenspaces of adjacency matrices of Hamming ball subgraphs, extending understanding of their spectral properties and maximal eigenvalues for large subsets of the Hamming cube.

Contribution

It provides a detailed spectral analysis of Hamming ball subgraphs and extends bounds on maximal eigenvalues for large subsets of the Hamming cube.

Findings

Eigenvalues and eigenspaces of Hamming ball adjacency matrices are explicitly described.

Hamming balls have essentially the largest maximal eigenvalue among subsets of the same size.

This property holds even for large subsets with subconstant fractions of the cube.

Abstract

We describe the eigenvalues and the eigenspaces of the adjacency matrices of subgraphs of the Hamming cube induced by Hamming balls, and more generally, by a union of adjacent concentric Hamming spheres. As a corollary, we extend the range of cardinalities of subsets of the Hamming cube for which Hamming balls have essentially the largest maximal eigenvalue (among all subsets of the same size). We show that this holds even when the sets in question are large, with cardinality which is an arbitrary subconstant fraction of the whole cube.