  The density of states, $\varrho(E)$, of graphene is investigated within the
tight binding (H\"uckel) approximation in the presence of vacancies. They
induce a non-vanishing density of zero modes, $n_\text{zm}$, that act as midgap
states: $\varrho(E)=n_\text{zm}\delta(E) + \text{smooth}$. As is well known,
the actual number of zero modes per sample can in principle exceed the
sublattice imbalance: $N_\text{zm}\geq |N_\text{A}-N_\text{B}|$, where
$N_\text{A}$, $N_\text{B}$ denote the number of carbon atoms in each
sublattice. In this work, we establish a stronger relation that is valid in the
thermodynamic limit and that involves the concentration of zero-modes:
$n_\text{zm}>|c_\text{A}-c_\text{B}|$, where $c_\text{A}$ and $c_\text{B}$
denote the concentration of vacancies per sublattice; in particular,
$n_\text{zm}$ is non-vanishing even in the case of balanced disorder,
$N_\text{A}/N_\text{B}=1$. Adopting terminology from benzoid graph theory, the
excess modes associated with the current carrying backbone (percolation
cluster) are called supernumerary. In the simplest cases such modes can be
associated with structural elements of internal boundaries like dangling bonds.
Our result suggest that the continuum limit of bipartite hopping models
supports nontrivial "supernumerary" terms that escape the present continuum
descriptions.
