On weighted spectral radius of unraveled balls and normalized Laplacian eigenvalues

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