  We study a class of graph Hamiltonians given by a type of quiver
representation to which we can associate (non)--commutative geometries. By
selecting gauging data these geometries are realized by matrices through an
explicit construction or a Kan-extension. We describe the changes in gauge via
the action of a regauging groupoid. It acts via matrices that give rise to a
noncommutative 2--cocycle and hence to a groupoid extension (gerbe).
  We furthermore show that automorphisms of the underlying graph of the quiver
can be lifted to extended symmetry groups of regaugings. In the commutative
case, we deduce that the extended symmetries act via a projective
representation. This yields isotypical decompositions and super--selection
rules.
  We apply these results to the PDG and honeycomb wire--networks using
representation theory for projective groups and show that all the degeneracies
in the spectra are consequences of these enhanced symmetries. This includes the
Dirac points of the G(yroid) and the honeycomb systems.
