On 2D Inverse Problems/Triangulations of surfaces

Let G be a graph embedded to a surface such that all faces of G are triangular. Such an embedding is called triangulation.

Medial graph of triangulation

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

$\sigma (G^{*})\mathrm{\setminus }\{6\}=\sigma (M(G))\mathrm{\setminus }\{6\}$

Medial graph