The existence of the paths in a graph and its dual with the following properties plays an important role in the connection between global and local properties of the graph: Two sets of paths between two subsets of boundary nodes in the graph and its dual are **Hamiltonian** if the paths go through every edge of the graph or its dual.

The following identity connects the weights of the paths of a network and its dual, an integral of conductivity over the network and the determinants of the Laplacians of the dual graphs, that admit Hamiltonian paths.

Let *G* be a graph embedded to a surface such that all faces of *G* are triangular. Such an embedding is called **triangulation**. The vertices of the dual graph *G ^{*}* of triangulation have degree

*3*.

**Exercise (***).**Generalize the following example to prove that the spectra of*G*and^{*}*M(G)*are equal, except possibly the eigenvalue*{6}*: