The Best Constant of Discrete Sobolev Inequality on the Regular Polyhedra including Double Bond

Abstract

Abstract. We have obtained the best constant of discrete Sobolev inequality on theregular polyhedra including double bond. Let N be the number of vertices. We introduce the discrete Laplacian A which is N × N real symmetric matrix. A has an eigenvalue 0 whose corresponding eigenspace is 1 dimension. If we introduce the pseudo Green matrix G∗, then G∗ is reproducing kernel by setting appropriate vector space and inner product. The maximum of the diagonal values of G∗ is the best constant of this inequality.