  We consider the Laplacian on a metric graph, equipped with Robin
($\delta$-type) vertex condition at some of the graph vertices and
Neumann-Kirchhoff condition at all others. The corresponding eigenvalues are
called Robin eigenvalues, whereas they are called Neumann eigenvalues if the
Neumann-Kirchhoff condition is imposed at all vertices. The sequence of
differences between these pairs of eigenvalues is called the Robin-Neumann gap.
We prove that the limiting mean value of this sequence exists and equals a
geometric quantity, analogous to the one obtained for planar domains. Moreover,
we show that the sequence is uniformly bounded and provide explicit upper and
lower bounds. We also study the possible accumulation points of the sequence
and relate those to the associated probability distribution of the gaps. To
prove our main results, we prove a local Weyl law, as well as explicit
expressions for the second moments of the eigenfunction scattering amplitudes.
