The best constant of discrete Sobolev inequality on the C60 fullerene buckyball

TL;DR

This paper determines the optimal constants for discrete Sobolev inequalities on the C60 fullerene buckyball by analyzing eigenvalues of the discrete Laplacian and computing Green matrices using Mathematica.

Contribution

It provides the exact eigenvalues of the discrete Laplacian on the buckyball and derives the best constants for the inequalities through Green matrix analysis.

Findings

Eigenvalues of the Laplacian are roots of degree 4 algebraic equations.

Green and pseudo Green matrices are explicitly computed.

Best constants are identified as diagonal entries of these matrices.

Abstract

The best constants of two kinds of discrete Sobolev inequalities on the C60 fullerene buckyball are obtained. All the eigenvalues of discrete Laplacian $A$ corresponding to the buckyball are found. They are roots of algebraic equation at most degree $4$ with integer coefficients. Green matrix $G(a)=(A+a I)^{-1}\ (0<a<\infty)$ and the pseudo Green matrix $G_*=A^{\dagger}$ are obtained by using computer software Mathematica. Diagonal values of $G_*$ and $G(a)$ are identical and they are equal to the best constants of discrete Sobolev inequalities.