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.