  Upper and lower estimates of eigenvalues of the Laplacian on a metric graph
have been established in 2017 by G. Berkolaiko, J.B. Kennedy, P. Kurasov and D.
Mugnolo. Both these estimates can be achieved at the same time only by highly
degenerate eigenvalues which we call maximally degenerate. By comparison with
the maximal eigenvalue multiplicity proved by I. Kac and V. Pivovarchik in 2011
we characterize the family of graphs exhibiting maximally degenerate
eigenvalues which we call lasso trees, namely graphs constructed from trees by
attaching lasso graphs to some of the vertices.
