  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.
