  We study the nodal count of the so-called bi-dendral graphs and show that it
exhibits an anomaly: the nodal surplus is never equal to 0 or $\beta$, the
first Betti number of the graph. According to the nodal-magnetic theorem, this
means that bands of the magnetic spectrum (dispersion relation) of such graphs
do not have maxima or minima at the "usual" symmetry points of the fundamental
domain of the reciprocal space of magnetic parameters.
  In search of the missing extrema we prove a necessary condition for a smooth
critical point to happen inside the reciprocal fundamental domain. Using this
condition, we identify the extrema as the singularities in the dispersion
relation of the maximal abelian cover of the graph (the honeycomb graph being
an important example).
  In particular, our results show that the anomalous nodal count is an
indication of the presence of the conical points in the dispersion relation of
the maximal universal cover. Also, we discover that the conical points are
present in the dispersion relation of graphs with much less symmetry than was
required in previous investigations.
