# On 2D Inverse Problems/The new spectral theorem

### Hamiltonian paths

The following identity connects the weights of the paths of a network and its dual, an integral of conductivity over the network and the eigenvalues of the Laplacian of the dual graphs, that admit Hamiltonian paths.

$\frac{\det(\Lambda(P,Q))}{\det(\Lambda^*(P^*,Q^*))} = \prod_{e\in E}\gamma(e)(\frac{\det(K^*)}{\det(K)}).$
Last modified on 16 January 2013, at 23:07