On the largest eigenvalue of a sparse random subgraph of the hypercube

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We consider a sparse random subraph of the n-cube where each edge appears independently with small probability p(n) =O(n⁻¹⁺ᵒ⁽¹⁾). In the most interesting regime when p(n) is not exponentially small we prove that the largest eigenvalue of the graph is asymtotically equal to the square root of the maximum degree.