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The best constant of three kinds of the discrete Sobolev inequalities on the complete graph. (English) Zbl 1310.46037
Summary: 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 \((d/dt + A + a_0 I) \cdots (d/dt + A + a_{M-1} I)\) with positive distinct characteristic roots \(a_{0}, \dots, a_{M-1}\). The second one is the difference operator \((A + a_{0} I) \cdots (A + a_{M-1} I)\) and the third one is the discrete polyharmonic operator \(A^M\). Here \(A\) is an \(N \times N\) real symmetric positive-semidefinite matrix whose eigenvector corresponding to zero eigenvalue is \(\mathbf 1 = ^t(1, 1,\dots, 1)\). A discrete heat kernel, a Green’s matrix and a pseudo Green’s matrix are obtained by means of \(A\).

46E39 Sobolev (and similar kinds of) spaces of functions of discrete variables
35K08 Heat kernel
35R02 PDEs on graphs and networks (ramified or polygonal spaces)
Full Text: DOI Euclid