Abstract
We introduce a discrete Laplacian A on the complete graph with N vertices, that is, KN. We obtain the best constants of three kinds of discrete Sobolev inequalities on KN. The background of the first inequality is the discrete heat operator (d/dt + A + a0I) ··· (d/dt + A + aM−1I) with positive distinct characteristic roots a0, ..., aM−1. The second one is the difference operator (A + a0I) ··· (A + aM−1I) and the third one is the discrete polyharmonic operator AM. Here A is an N × N real symmetric positive-semidefinite matrix whose eigenvector corresponding to zero eigenvalue is 1 = t(1, 1, ..., 1). A discrete heat kernel, a Green's matrix and a pseudo Green's matrix are obtained by means of A.
