  Deriving mathematical expressions of two zero modes for a $\pi$-band
tight-binding model, we identify a class of bipartite graphs having the same
number of subgraph sites, where each graph represents one of the
quasi-hexagonal nanographene molecule with a center vacancy (VANG). Indeed, in
a VANG molecule, C$_{60}$H$_{24}$, showing stability in a density-functional
simulation at the highest occupied level, there appear two pseudo-degenerate
zero modes, a vacancy-centered quasi-localized zero mode, and extending zero
mode with a $\sqrt{3} \times \sqrt{3}$ structure. Since there is a finite
energy gap between these two zero-energy modes and the other modes, low-lying
states composed of quasi-degenerate zero modes appear as magnetic multiplets.
Thus, the unique magnetic characteristics derived in our theory are expected to
hold for synthesized VANG molecules in reality.
