  Eigenvalue behavior for the equation -\lambda y"=Vu on the edges of a graph G
of final total length, with a non-negative weight function V and under the
Kirchhoff matching conditions at the vertices and zero boundary condition at at
least one point of G, is studied. It is shown that the eigenvalues satisfy an
inequality which involves the length |G| and the total mass corresponding to V
but otherwise does not depend on the graph. Applications and generalizations of
this result are also discussed.
