  We consider a family of non-compact manifolds $X_\eps$ (``graph-like
manifolds'') approaching a metric graph $X_0$ and establish convergence results
of the related natural operators, namely the (Neumann) Laplacian $\laplacian
{X_\eps}$ and the generalised Neumann (Kirchhoff) Laplacian $\laplacian {X_0}$
on the metric graph. In particular, we show the norm convergence of the
resolvents, spectral projections and eigenfunctions. As a consequence, the
essential and the discrete spectrum converge as well. Neither the manifolds nor
the metric graph need to be compact, we only need some natural uniformity
assumptions. We provide examples of manifolds having spectral gaps in the
essential spectrum, discrete eigenvalues in the gaps or even manifolds
approaching a fractal spectrum. The convergence results will be given in a
completely abstract setting dealing with operators acting in different spaces,
applicable also in other geometric situations.
