Alon–Boppana-Type Bounds for Weighted Graphs

The unraveled ball of radius r centered at a vertex v in a weighted graph G is the ball of radius r centered at v in the universal cover of G. We present a general bound on the maximum spectral radius of unraveled balls of fixed radius in a weighted graph. The weighted degree of a vertex in a weighted graph is the sum of weights of edges incident to the vertex. A weighted graph is called regular if the weighted degrees of its vertices are the same. Using the result on unraveled balls, we prove a variation of the Alon–Boppana theorem for regular weighted graphs.