  We address the energy spectrum of honeycomb lattice with various defects or
impurities under a perpendicular magnetic field. We use a tight-binding
Hamiltonian including interactions with the nearest neighbors and investigate
its energy structure for two different choices of point defects or impurities.
In the first case, we fix a unit cell consisting of 8 lattice points and survey
the energy eigenvalues in the presence of up to 2 point defects. Then it turns
out that the existence of the fractal energy structure, called Hofstadter's
butterfly, depends on the choice of defect pairs. In the second case, we extend
the size of a unit cell which contains a single point defect and up to 32
lattice points. The fractal structures indeed appear for those cases and there
exist a robust gapless point in the $E=0$ eV line without depending on both the
size of unit cells and the shape of lattices. Therefore we keep an immortal
butterfly since such a robust point corresponds to the center of a butterfly.
Consequently we predict that the presence of a butterfly in a graph is
equivalent to that of fractality.
