Introduction:
In a two-dimensional sheet of graphene, the low-energy electronic excitations are described by a linear dispersion relation, given by E(k) = ħvk, where v is the velocity of the excitations. The density of states (DOS) refers to the number of available electronic states per unit energy range at a given energy. In this case, we need to determine the functional form of the DOS and discuss its proportionality.

Explanation:
To understand the density of states in graphene, we can consider a two-dimensional Brillouin zone (BZ) that characterizes the electronic structure of the material. The BZ is a representation of the possible momentum states of the electrons in the system.

1. Derivation of DOS:
- The number of states in the BZ is given by the area enclosed by the BZ and is denoted as N(k), where k represents the momentum vector.
- The energy associated with each momentum state is given by the dispersion relation E(k) = ħvk.
- The density of states (DOS) can be defined as the derivative of the number of states with respect to energy, i.e., DOS(E) = dN(k)/dE.
- Since the number of states N(k) is proportional to the area of the BZ, we can write dN(k) = A(k)dk, where A(k) is the area of the BZ in k-space.
- Substituting the expression for dN(k) into the definition of DOS, we have DOS(E) = dN(k)/dE = A(k)dk/dE = A(k)/v.
- Since A(k) is a constant for a given material, the DOS(E) is proportional to 1/v.

2. Proportionality of DOS:
The velocity v in the dispersion relation E(k) = ħvk characterizes the behavior of the low-energy electronic excitations in graphene. It is found that the velocity of the excitations in graphene is a constant, independent of the energy or momentum of the electrons.
- This constant velocity implies that the density of states in graphene is proportional to 1/v, where v is the velocity of the excitations.
- As a result, the density of states in graphene increases linearly with energy, indicating a linear increase in the number of available electronic states per unit energy range.

Conclusion:
The density of states in a two-dimensional sheet of graphene is proportional to 1/v, where v represents the velocity of the low-energy electronic excitations. This proportionality arises from the linear dispersion relation E(k) = ħvk, which characterizes the behavior of the excitations in graphene. The constant velocity of the excitations in graphene leads to a linear increase in the density of states with energy, indicating a linear increase in the number of available electronic states per unit energy range.