Lamplighter groups, de Brujin graphs, spider-web graphs and their spectra

Abstract

We study the infinite family of spider-web graphs , , and , initiated in the 50s in the context of network theory. It was later shown in physical literature that these graphs have remarkable percolation and spectral properties. We provide a mathematical explanation of these properties by putting the spider-web graphs in the context of group theory and algebraic graph theory. Namely, we realize them as tensor products of the well-known de Bruijn graphs with cyclic graphs and show that these graphs are described by the action of the lamplighter group on the infinite binary tree. Our main result is the identification of the infinite limit of , as , with the Cayley graph of the lamplighter group which, in turn, is one of the famous Diestel–Leader graphs . As an application we compute the spectra of all spider-web graphs and show their convergence to the discrete spectral distribution associated with the Laplacian on the lamplighter group.