  Studying the tight binding model in an applied rational magnetic field (H) we
show that in graphene there are very unusual Landau levels situated in the
immediate vicinity of the saddle point (M-point) energy epsilon_M. Landau
levels around $\epsilon_M$ are broadened into minibands (even in relatively
weak magnetic fields ~40-53 T) with the maximal width reaching 0.4-0.5 of the
energy separation between two neighboring Landau levels though at all other
energies the width of Landau levels is practically zero. In terms of the
semiclassical approach a broad Landau level or magnetic miniband at epsilon_M
is a manifestation of the so called self-intersecting orbit signifying an
abrupt transition from the semiclassical trajectories enclosing the $\Gamma$
point to the trajectories enclosing the K point in the momentum space. The
structure of broad magnetic minibands is clearly fractal demonstrating fast
oscillations of the band energy E(k) with the magnetic wave vector k.
Remarkably, the saddle point virtually does not affect the diamagnetic response
of graphene, which is caused mostly by electron states in the vicinity of the
Fermi energy \epsilon_F. Experimentally, the effect of the broading of Landau
levels can possibly be observed in twisted graphene where two saddle point
singularities can be brought close to the Fermi energy.
