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

### Christophe Texier 1, 2

#### Journal of Physics A General Physics 41 (2008) 085207

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 $S_\mathcal{G}(\gamma)=\det(\gamma-\Delta)$ 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 Schrödinger operator $\det(\gamma-\Delta+V(x))$.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI - Paris Sud
• 2. Laboratoire de Physique des Solides (LPS),
CNRS : UMR8502 – Université Paris XI - Paris Sud