2 dots since the number of carbon atoms increases rapidly with the size. We have found a scaling result from which one can infer results for larger dots from those of smaller dots. For different rectangular-shaped graphene dots and values of φ it can be described well by the following di- mensionless form where ℓ = 3 / $^{4}$a 0 (4 πφ ) 1 / 2 is the magnetic length and f ( x ) is a scaling function, see Fig.2. The total number of hexagons in the dot is N$_{h}$ = L$_{x}$L$_{y}$/A$_{h}$ . Our numerical result shows that the dependence of N$_{D}$ ( φ ) on φ is initially non-linear in the regime where the diameter of the cyclotron motion is comparable to the system length, 2 ℓ ∼ L$_{x,y}$ . II. NUMBER OF STATES IN THE QUASI-DEGENERATE LEVEL Our Hamiltonian is N$_{D}$ ( φ ) = L$_{x}$L$_{y}$ A$_{h}$ f ( ℓ 2 a$_{0}$ ( L$_{x}$ + L$_{y}$ ) ) , (3) H = − ∑ <i,j> t$_{ij}$$_{c}$† $_{i}$c$_{j}$ , (4) where t$_{ij}$ = te i e ℏ ∫ ⃗ R i ⃗ R j ⃗ $^{A}$·d⃗r are the hopping parameters and c + i creates an electron at site i . Here we use a Lan- dau gauge ⃗ A = B ( − y, 0 , 0). The summation < i, j > is over nearest neighbor sites and t = 2 . 5eV. The eigen- state with eigenenergy ϵ$_{n}$ is denoted by φ$_{ϵ}$$_{n}$ ( ⃗ R ), where ⃗ R labels each lattice point. Because of electron-hole sym- metry eigenvalues appear in pairs of positive and nega- tive values ( ϵ, − ϵ ), and the probability wavefunctions of a pair of states, ( | φ$_{ϵ}$$_{n}$ ( ⃗ R ) | $^{2}$, | $^{φ}$− $_{ϵ$_{n}$}$( ⃗ R ) | $^{2}$), are identical. Our numerical results are consistent with this. Figs.3(a) and (b) display the energy spectra near zero energy for φ = 0 and 0 . 01. At φ = 0 there are approx- imately 20 states within | ϵ$_{n}$ | < 0 . 01 eV , consistent with the analytical result of Eq.(1). At φ = 0 . 01 the numerical value is increased to 24. Fig.4(a) shows how some energy levels ϵ$_{n}$ at φ = 0 change as a function of the magnetic flux φ . These energy levels do not anticross. We ob- serve that nearly zero energies at φ = 0 do not change noticeably in magnitude as φ varies. There are N$_{l}$ such localized surface states. On the other hand, we find that as φ increases non-zero energies become smaller and move closer to zero. This implies that, for a given energy inter- val δ , the number of states in it, N$_{T}$ ( φ ), increases with φ . From Fig.4(b) we see that it displays a non-linear depen- dence on φ . For a large dot of size 50 × 50 nm 2 a similar dependence of N$_{T}$ on B is seen, as shown in Fig.5. Non- linear dependence occurs in the regime 2 ℓ/L$_{x}$ ∼ 0 . 5. As a test of our numerical procedures we have verified that the sum of N$_{D}$ ( φ ) = N$_{T}$ ( φ ) − N$_{l}$ and the number of mag- netic edge states is equal to the total bulk Landau level degeneracy 2 D$_{B}$ ( D$_{B}$ = 2 L$_{x}$L$_{y}$ $_{3}$√$_{3}$$_{a}$ 2 0 φ is the degeneracy per valley). Our numerical results indicate that for relatively small rectangular-shaped graphene dots with size less than of order 10 2 ˚ A the number of states at the Fermi level display a negligible magnetic field dependence for values that are usually accessible experimentally ( B < 10 T corresponds, in dimensionless magnetic flux, to φ = Φ Φ 0 < 10 − $^{4}$, where Φ$_{0}$ = hc/e and Φ = BA$_{h}$ with the area of a hexagon A$_{h}$ = $_{3}$√$_{3}$ 2 a 2 $_{0}$). On the other hand, in larger dots this dependence is significant. However, it is computationally difficult to investigate large rectangular-shaped graphene (the energy δ is typically less than the quantization en- ergy of a rectangular dot, which can be estimated using the Dirac equation: γk$_{min}$ ∼ t a $_{L$_{x,y}$}$, where γ = √$_{3}$$_{ta/}$$_{2}$ with the hopping energy t ). Here magnetic edge states are not included since their energies are larger than δ . N$_{D}$ ( φ ) is the number states at zero magnetic field that merge into the energy interval δ around zero energy as the dimensionless magnetic flux φ = Φ Φ 0 increases. This effect provides a means to control the number of states at the Fermi level. There are other states with nearly zero energies at φ = 0, which remain so even at φ = 0. The localization lengths of these states are shorter than the magnetic length. We denote the number of these states by N$_{l}$ . N$_{T}$ ( φ ) = N$_{l}$ + N$_{D}$ ( φ ) (2) Our study shows that the number of states within the energy interval δ around zero energy is given by change in the presence of a magnetic field in the regime where Hofstadter-butterfly effects[18, 19, 21] are negligi- ble. In neutral graphene dots the Fermi level has zero energy, and, consequently, magnetic, optical, and STM properties are expected to depend on the number of avail- able states near zero energy. Our investigation shows that the wavefunctions of nearly zero energy states are all peaked on the zigzag edges with or without appre- ciable weight inside the dot. This may be understood as mixing of LLL and surface states by armchair edges of the square dot through intervalley scattering, which is unique to the square dot (This will be explained in Sec.IV). FIG. 2: Number of nearly zero energy states that are induced by a magnetic field follow a scaling curve when plotted as a function of 1 ( M $_{+}$√$_{3}$$_{N}$$_{+2}$$_{/}$ √$_{3)}$$_{φ}$ for different rectangular shaped graphene dots. Here δ = 0 . 01 eV .