An edge-separating theorem on the second smallest normalized Laplacian eigenvalue of a graph and its applications

Let λ2(G)λ2(G) be the second smallest normalized Laplacian eigenvalue of a graph GG. In this paper, we investigate the behavior on λ2(G)λ2(G) when the graph GG is perturbed by separating an edge. This result can be used to determine all trees and unicyclic graphs with λ2(G)≥1−22. Moreover, the trees and unicyclic graphs with λ2(G)=1−22 are also determined, respectively.