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.

The relatively new weighted voting theory applies to many important organizations such as the United States Electoral College and the International Monetary Fund. Various power indexes are used to establish a relationship between weights and influence; in 1965, the Banzhaf Power Index was used to show that areas of Nassau County were unrepresented in the county legislature. It is of interest to...