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)}). $$