  Quantum graphs were introduced to model free electrons in organic molecules
using a self-adjoint Hamiltonian on a network of intervals. A second graph
quantization describes wave propagation on a graph by specifying scattering
matrices at the vertices. A question that is frequently raised is the extent to
which these models are the same or complementary. In particular, are all energy
independent unitary vertex scattering matrices associated with a self-adjoint
Hamiltonian? Here we review results related to this issue. In addition, we
observe that a self-adjoint Dirac operator with four component spinors produces
a secular equation for the graph spectrum that matches the secular equation
associated with wave propagation on the graph when the Dirac operator describes
particles with zero mass and the vertex conditions do not allow spin rotation
at the vertices.
