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THE JACOBIAN DETERMINANT OF THE CONDUCTIVITIES-TO-RESPONSE-MATRIX MAP FOR WELL-CONNECTED CRITICAL CIRCULAR PLANAR GRAPHS

We consider the map from conductivities to the response matrix. For critical circular planar graphs, this map is known to be invertible, at least when the conductivities are positive. We calculate the Jacobian determinant of this map, which turns out to have a fairly simple form. Using this we show that for arbitrary critical circular planar networks, the map from conductivities to the response matrix is generally invertible when the conductivities are allowed to be negative or complex (but nonzero). This is an alternate proof to a result in [4].