Eigenvalues of subgraphs of the cube

TL;DR

This paper investigates the maximum eigenvalues of subgraphs of hypercubes, providing evidence that Hamming balls are optimal, especially for large dimensions, using compression techniques.

Contribution

It proves that Hamming balls of fixed radius maximize the largest eigenvalue exactly for large dimensions, supporting the conjecture that they are extremal.

Findings

Hamming balls of radius o(d) have eigenvalues close to the maximum

Hamming balls with fixed radius maximize eigenvalues exactly for large d

Compression methods are used to establish these results

Abstract

We consider the problem of maximising the largest eigenvalue of subgraphs of the hypercube $Q_d$ of a given order. We believe that in most cases, Hamming balls are maximisers, and our results support this belief. We show that the Hamming balls of radius $o(d)$ have largest eigenvalue that is within $1 + o(1)$ of the maximum value. We also prove that Hamming balls with fixed radius maximise the largest eigenvalue exactly, rather than asymptotically, when $d$ is sufficiently large. Our proofs rely on the method of compressions.