On weighted spectral radius of unraveled balls and normalized Laplacian eigenvalues

For a graph G, the unraveled ball of radius r centered at vertex v is in universal cover G. We obtain lower bound on weighted spectral balls fixed with positive weights edges, which used to present an upper sth (where s≥2) smallest normalized Laplacian eigenvalue irregular graphs under minor assumptions. Moreover, when s=2, result may be regarded as Alon–Boppana type for class graphs.