On 2D Inverse Problems/Triangulations of surfaces

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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^*)\backslash \{6\}=\sigma(M(G))\backslash \{6\}
Medial graph