Braess's paradox for the spectral gap in random graphs and delocalization of eigenvectors

We study how the spectral gap of the normalized Laplacian of a random graph changes when an edge is added to or removed from the graph. There are known examples of graphs where, perhaps counterintuitively, adding an edge can decrease the spectral gap, a phenomenon that is analogous to Braess's paradox in traffic networks. We show that this is often the case in random graphs in a strong sense. More precisely, we show that for typical instances of Erd\H{o}s-R\'enyi random graphs $G(n,p)$ with constant edge density $p \in (0,1)$, the addition of a random edge will decrease the spectral gap with positive probability, strictly bounded away from zero. To do this, we prove a new delocalization result for eigenvectors of the Laplacian of $G(n,p)$, which might be of independent interest.