  The Neumann points of an eigenfunction $f$ on a quantum (metric) graph are
the interior zeros of $f'$. The Neumann domains of $f$ are the sub-graphs
bounded by the Neumann points. Neumann points and Neumann domains are the
counterparts of the well-studied nodal points and nodal domains.
  We prove some foundational results in this field: bounds on the number of
Neumann points and properties of the probability distribution of this number.
Two basic properties of Neumann domains are presented: the wavelength capacity
and the spectral position. We state and prove bounds on those as well as key
features of their probability distributions.
  To rigorously investigate those probabilities, we establish the notion of
random variables for quantum graphs. In particular, we provide conditions for
considering spectral functions of quantum graphs as random variables with
respect to the natural density on $\mathbb{N}$.
