TABLE I.

The total number of sites $N=7\left[{\left(\frac{3+\sqrt{5}}{2}\right)}^{\ell }+{\left(\frac{3-\sqrt{5}}{2}\right)}^{\ell }-2\right]$ grows exponentially as a function of the number of rings ℓ. Each finite graph is mapped onto a continuous disk of radius $L=\sqrt{N/\left(N+28\right)}$. The ground state energy E₀ of the hopping Hamiltonian (1), defined as the lowest eigenvalue of the matrix H = −A, can be estimated from the lowest eigenvalue of Δ_g on the finite disk of radius L < 1, which gives $E_0^{\left(\text{cont}\right)}$. Both values agree excellently for sufficiently large ℓ; see also Fig. 3. For ℓ ⩾ 8 we have to resort to less precise sparse matrix methods to estimate E₀.

	Number of rings ℓ
	1	2	3	4	5	6	7	8	9	10
Number of graph sites N	7	35	112	315	847	2240	5887	15435	40432	105875
Effective disk radius L	0.447	0.745	0.894	0.958	0.984	0.994	0.998	0.9990	0.9997	0.9999
Ground state energy (graph) E0	−2	−2.636	−2.787	−2.847	−2.877	−2.894	−2.905	−2.91	−2.92	−2.92
Ground state energy (continuum) $E_0^{(\mathrm{cont})}$	−1.500	−2.570	−2.770	−2.842	−2.876	−2.895	−2.906	−2.914	−2.920	−2.924