### Hamiltonian pathsEdit

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.

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.

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