Weighted expanders and the anisotropic Alon–Boppana theorem

We present the anisotropic version of the classical Alon–Boppana theorem on the asymptotic spectrum of random walks on infinite families of graphs. The relations to the Grigorchuk–Żuk theory—on the space of graphs with uniformly bounded degree and the continuity of the spectral measure of Markov operators related to the Alon–Boppana theorem and the Burger–Greenberg–Serre theorem—are also indicated.