Eigenvalues of subgraphs of the cube

We consider the problem of maximising the largest eigenvalue of subgraphs of the hypercube Qd 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.