Electrical Network Model and Reducibility

We can think of the graph energies as the energies of an electrical network with resistors for edges and voltages given by the values of our function on the graph. Up to a constant, this is physically correct. We can do the standard transformations of resistors to determine effective resistance between any two nodes. One transformation of particular importance is the `Delta-Y' transformation, which takes a triangular network, adds a vertex, and then replaces the edges from the triangle by edges going to the new central vertex. There are simple invertible formulae giving the new resistances in terms of the old ones. Using this transformation, we transform a level one gasket with resistances all equal to 1 into a level zero gasket and the new effective resistances correspond to the reciprocal of the correct renormalization factor.

We would like to be able to do this for higher order gaskets. Unfortunately, the number of initial edges is n(n-1)/2 but the number of vertices is only n. This means that the parameter space for the 'Y' has much lower dimension in general than the 'Delta'. We call 'Delta' networks which have a corresponding 'Y' network preserving effective resistances a reducible network. We derived a simple reducibility criterion for SG4. We require that all sums R(i,j)+R(k,l) for i, j, k, and l all distinct be the same, where R(i,j) is the effective resistance between nodes i and j in the 'Delta' network. This reduces to a very simple relation between the initial conductances. Applying this formula, we found that if we assume a reducible network is renormalizable then it must have constance conductances, so reducibility doesn't help us move beyond the trivial case for SG4. At this point, we stopped trying to derive reducibility conditions for higher order gaskets, as they were both computationally cumbersome and seemed less likely to produce non-trivial results.

Reducible electrical networks proved unhelpful for the simple reason that for n>3 we have (n-1)/2 not equal to 1.

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