Balram, Ajit C. ; Dhar, Deepak (2011) Vibrational spectrum of spiderweb networks Arxiv  eprints . pp. 113.

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Official URL: http://arxiv.org/pdf/1111.0741v1.pdf
Abstract
We consider the vibrational spectrum of masses connected by springs forming a spiderweb 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 mth 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 δfunctions, 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.
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Deposited On:  10 Feb 2012 04:19 
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