Vibrational Spectrum of Spider Networks

A 2^N x M - node spider network is a set of M-layers each of which has 2^N nodes which can be represented by N-bit binary numbers. Each site in a layer is connected to two sites in the next layer by a set of rules and this procedure is continued for the next layer and so on to generate the entire set of connections in the network. The last layer is connected back to the first layer. An example of a 4-layer, 2-bit network is shown in the attached figure. (Rule: ab->b0(colored red) and ab->b1(colored green) where a and b are either 0 or 1).

Consider an atom of mass M placed at each node. Each site is connected by a spring of spring constant K to the atoms to which it has connections. In the talk I will find the normal modes of this lattice.