On 2D Inverse Problems/The new spectral theorem

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Hamiltonian paths[edit]

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)}).