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.

${\frac {\det(\Lambda (P,Q))}{\det(\Lambda ^{*}(P^{*},Q^{*}))}}=\prod _{e\in E}\gamma (e)({\frac {\det(K^{*})}{\det(K)}}).$