  The spectral gap for Laplace operators on metric graphs is investigated in
relation to graph's connectivity, in particular what happens if an edge is
added to (or deleted from) a graph. It is shown that in contrast to discrete
graphs connection between the connectivity and the spectral gap is not
one-to-one. The size of the spectral gap depends not only on the topology of
the metric graph but on its geometric properties as well. It is shown that
adding sufficiently large edges as well as cutting away sufficiently small
edges leads to a decrease of the spectral gap. Corresponding explicit criteria
are given.
