On the spectrum of the Laplace operator of metric graphs attached at a vertex -- Spectral determinant approach

We consider a metric graph $\mathcal{G}$ made of two graphs ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ attached at one point. We derive a formula relating the spectral determinant of the Laplace operator $S_{\mathcal{G}}(\gamma )=\det (\gamma -\mathrm{\Delta })$ in terms of the spectral determinants of the two subgraphs. The result is generalized to describe the attachment of $n$ graphs. The formulae are also valid for the spectral determinant of the Schr"odinger operator $\det (\gamma -\mathrm{\Delta }+V(x))$.