Sharp spectral bounds of several graph parameters using eigenvector norms

We investigate the role of the 1- and ∞-norms of eigenvectors in spectral graph theory. In particular, we produce several randomized algorithms which show that various graph-theoretic parameters can be tightly bounded by the eigenvalues as well as norms of the corresponding eigenvectors. Further, in some cases, these inequalities can determine the parameters exactly. Our results include: a spectral bound for the densest subgraph problem, an adapted “converse” to the Expander Mixing Lemma, and an adapted spectral approach to the maximum cut problem.