Discrete Quantitative Nodal Theorem

We prove a theorem that can be thought of as common generalization the Discrete Nodal Theorem and (one direction of) Cheeger's Inequality for graphs. A special case this result will assert if second third eigenvalues Laplacian are at least $\varepsilon$ apart, then subgraphs induced by positive negative supports eigenvector belonging to $\lambda_2$ not only connected, but edge-expanders (in weighted sense, with expansion depending on $\varepsilon$).