  We study the transmission of a quantum particle along a straight
input--output line to which a graph $\Gamma$ is attached at a point. In the
point of contact we impose a singularity represented by a certain properly
chosen scale-invariant coupling with a coupling parameter $\alpha$. We show
that the probability of transmission along the line as a function of the
particle energy tends to the indicator function of the energy spectrum of
$\Gamma$ as $\alpha\to\infty$. This effect can be used for a spectral analysis
of the given graph $\Gamma$. Its applications include a control of a
transmission along the line and spectral filtering. The result is illustrated
with an example where $\Gamma$ is a loop exposed to a magnetic field. Two more
quantum devices are designed using other special scale-invariant vertex
couplings. They can serve as a band-stop filter and as a spectral separator,
respectively.
