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.

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