  It has been suggested that the distribution of the suitably normalized number
of zeros of Laplacian eigenfunctions contains information about the geometry of
the underlying domain. We study this distribution (more precisely, the
distribution of the "nodal surplus") for Laplacian eigenfunctions of a metric
graph. The existence of the distribution is established, along with its
symmetry. One consequence of the symmetry is that the graph's first Betti
number can be recovered as twice the average nodal surplus of its
eigenfunctions. Furthermore, for graphs with disjoint cycles it is proven that
the distribution has a universal form --- it is binomial over the allowed range
of values of the surplus. To prove the latter result, we introduce the notion
of a local nodal surplus and study its symmetry and dependence properties,
establishing that the local nodal surpluses of disjoint cycles behave like
independent Bernoulli variables.
