There are several reasons why the study of the spectrum of the Laplacian in a narrow neighborhood of an embedded graph is interesting. The graph can be embedded into a Euclidean space or it can be embedded into a manifold. In his pioneering work [3], Colin de Verdière

Abstract. Let (Gε)ε>0 be a family of ’ε-thin ’ Riemannian manifolds modeled on a finite metric graph G, for example, the ε-neighborhood of an embedding of G in some Euclidean space with straight edges. We study the asymptotic behavior of the spectrum of the Laplace-Beltrami operator on Gε as ε → 0, for various boundary conditions. We obtain complete asymptotic expansions for the kth eigenvalue and the eigenfunctions, uniformly for k ≤ Cε −1, in terms of scattering data on a non-compact limit space. We then use this to determine the quantum graph which is to be regarded as the limit object, in a spectral sense, of the family (Gε). Our method is a direct construction of approximate eigenfunctions from the scattering and graph data, and use of a priori estimates to show that all

propagation in periodic networks of thin fibers

Abstract. Quantum networks are often modelled using Schrödinger operators on metric graphs. To give meaning to such models one has to know how to interpret the boundary conditions which match the wave functions at the graph vertices. In this article we give a survey, technically not too heavy, of several recent results which serve this purpose. Specifically, we consider approximations by means of “fat graphs ” — in other words, suitable families of shrinking manifolds — and discuss convergence of the spectra and resonances in such a setting.