Eigenvalues of subgraphs of the cube
B. Bollobás, J. Lee, S. Letzter
European Journal of Combinatorics 70, 125 (2018)
#graphtheory#combinatorics#spectraltheory
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..
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