### Abstract

We investigate the behavior of the eigenvalues of the Laplacian, or a similar operator, on a family of Riemannian manifolds with boundary, called fat graphs, obtained by associating to the edges of a given finite graph cross-sectional Riemannian manifolds with boundary, and also to the vertices certain Riemannian manifolds with boundary, glueing them according to the graph structure, and scaling them by a factor of $\varepsilon$ while keeping the lengths of the edges fixed. The simplest model of this is the $\varepsilon$-neighborhood of a the graph embedded with straight edges in $R^n$.

We determine the asymptotics of the eigenvalues with various boundary conditions as $\varepsilon\to 0$ in terms of combinatorial and scattering data.