$\frac{\det\Lambda(P,Q)}{\det\Lambda^*(P^*,Q^*)} = \prod_{e\in E}\gamma(e)\frac{\det(C^*)}{\det(C)}.$
Exercise (***). Generalize the following example to prove that the spectra of G* and M(G) are equal, except possibly the eigenvalue {6}: $\sigma(G^*)\cup \{6\}=\sigma(M(G))\cup \{6\}$