The best constant of three kinds of the discrete Sobolev inequalities on the complete graph

Abstract

We introduce a discrete Laplacian A on the complete graph with N vertices, that is, K_N. We obtain the best constants of three kinds of discrete Sobolev inequalities on K_N. The background of the first inequality is the discrete heat operator (d/dt + A + a₀I) ··· (d/dt + A + a_M−1I) with positive distinct characteristic roots a₀, ..., a_M−1. The second one is the difference operator (A + a₀I) ··· (A + a_M−1I) and the third one is the discrete polyharmonic operator A^M. 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.