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The Spectral Gap of a Random Subgraph of a Graph

We examine the relationship of a graph G and its random subgraphs, which are defined by independently choosing each edge with probability p. Suppose that G has a spectral gap λ (in terms of its normalized Laplacian) and minimum degree d min. Then we can show that a random subgraph of G on n vertices with edge-selection probability p almost surely has as its spectral gap .