  In this thesis, we study Laplacian eigenfunctions on metric graphs, also
known as quantum graphs. We restrict the discussion to standard quantum graphs.
These are finite connected metric graphs with functions that satisfy Neumann
vertex conditions.
  The first goal of this thesis is the study of the nodal count problem. That
is the number of points on which the $n$th eigenfunction vanishes. We provide a
probabilistic setting using which we are able to define the nodal
count\textquoteright s statistics. We show that the nodal count statistics
admit a topological symmetry by which the first Betti number of the graph can
be obtained. We revise a conjecture that predicts a universal Gaussian behavior
of the nodal count statistics for large graphs and we prove it for a certain
family of graphs.
  The second goal is to formulate and study the Neumann count, which is the
number of local extrema of the $n$th eigenfunction. This counting problem is
motivated by the Neumann partitions of planar domains, a novel concept in
spectral geometry. We provide uniform bounds on the Neumann count and
investigate its statistics. We show that the Neumann count provides
complimentary geometrical information to that obtained from the nodal count. We
show that for a certain family of growing tree graphs the Neumann count
statistics approaches a Gaussian distribution.
  The third goal is a genericity result, which justifies the generality of the
Neumann count discussion. To this day it was known that generically,
eigenfunctions do not vanish on vertices. We generalize this result to
derivatives at vertices as well. That is, generically, the derivatives of an
eigenfunction on interior vertices do not vanish.
