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Vibrational spectrum of spider-web networks

We consider the vibrational spectrum of masses connected by springs forming a spider-web network. These graphs were first introduced in connection with the design of telephone switching networks. We consider a graph consisting of M levels, each of which has q N vertices. Each vertex in the m-th level is connected to q vertices each in the (m±1)th levels. We use the large symmetry group of the graph to determine all normal models and their frequencies. In the limit of large M and N, the spectrum is a set of δ-functions, and almost all the modes are localized. The fractional number of modes with frequency less than ω varies as exp (-C/ω) for ω tending to zero, where C is a constant. This implies that for an unbiased random walk on the vertices of this graph, the probability of return to the origin at time t varies as exp(- C' t 1/3 ), for large t, where C' is a constant.