The best constant of discrete Sobolev inequality on the C60 Fullerene Buckyball

The best constants of two types of discrete Sobolev inequality for the C60 Fullerene Buckyball are obtained. We first introduce the discrete Laplacian A corresponding to the buckyball and find the Green matrix G(a) = (A + aI)⁻¹ (0 < a < ∞) and pseudo Green matrix G_* = A^†, which is a Penrose–Moore generalized inverse matrix of A. These matrices G_* and G(a) are reproducing kernels of suitable Hilbert spaces. From the reproducing relations, we derive discrete Sobolev inequalities, which estimate the deviation of atoms from above by certain energy norms. The diagonal values of G_* and G(a) are identical and equal to the best constants of discrete Sobolev inequalities. (author)