The Spectral Gap of a Random Subgraph of a Graph

Abstract

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 $\lambda$ (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 $\lambda - O\big(\sqrt{\frac{\log n }{pd_{\min}}}+\frac{(\log n)^{3/2}}{pd_{\min}(\log \log n)^{3/2}}\big)$.