Spider web networks

Abstract

A 2^N\times 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. A site in layer r may be represented by an N-bit integer. The site in the r^{th} layer with the bit label abcd... ij is connected to two sites with labels bcd...ij0 and bcd...ij1 in the layer r+1. The layer M is connected in the same way to layer 1. We enumerate self-avoiding walks(SAWs) and self-avoiding polygons(SAPs) on a spider-web network. The high-temperature expansion of the Ising model on this network is discussed. Then we consider a spring network on this graph, with a mass m placed at each node, and for each link, there is a spring of spring constant K connecting the masses at the ends. I discuss the normal modes of this lattice.