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

Abstract

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.